CINXE.COM
Total variation - Wikipedia
<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Total variation - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"013d925f-6955-4947-bc01-66c6e9d616f9","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Total_variation","wgTitle":"Total variation","wgCurRevisionId":1268508491,"wgRevisionId":1268508491,"wgArticleId":683561,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Articles lacking reliable references from February 2012","All articles lacking reliable references","Articles needing additional references from September 2022","All articles needing additional references","Articles needing additional references from May 2012","Articles lacking in-text citations from February 2012","All articles lacking in-text citations","CS1 Italian-language sources (it)", "CS1 French-language sources (fr)","CS1 German-language sources (de)","Pages that use a deprecated format of the math tags","Mathematical analysis"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Total_variation","wgRelevantArticleId":683561,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false, "wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1936288","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site", "mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.16"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Total variation - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Total_variation"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Total_variation&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Total_variation"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Total_variation rootpage-Total_variation skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Total+variation" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Total+variation" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Total+variation" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Total+variation" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Historical_note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historical_note"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historical note</span> </div> </a> <ul id="toc-Historical_note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definitions</span> </div> </a> <button aria-controls="toc-Definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definitions subsection</span> </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-Total_variation_for_functions_of_one_real_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_variation_for_functions_of_one_real_variable"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Total variation for functions of one real variable</span> </div> </a> <ul id="toc-Total_variation_for_functions_of_one_real_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_variation_for_functions_of_n_>_1_real_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_variation_for_functions_of_n_>_1_real_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Total variation for functions of <i>n</i> > 1 real variables</span> </div> </a> <ul id="toc-Total_variation_for_functions_of_n_>_1_real_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_variation_in_measure_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_variation_in_measure_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Total variation in measure theory</span> </div> </a> <ul id="toc-Total_variation_in_measure_theory-sublist" class="vector-toc-list"> <li id="toc-Classical_total_variation_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Classical_total_variation_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Classical total variation definition</span> </div> </a> <ul id="toc-Classical_total_variation_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_definition_of_total_variation_norm" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modern_definition_of_total_variation_norm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Modern definition of total variation norm</span> </div> </a> <ul id="toc-Modern_definition_of_total_variation_norm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_variation_norm_of_complex_measures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Total_variation_norm_of_complex_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Total variation norm of complex measures</span> </div> </a> <ul id="toc-Total_variation_norm_of_complex_measures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_variation_norm_of_vector-valued_measures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Total_variation_norm_of_vector-valued_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.4</span> <span>Total variation norm of vector-valued measures</span> </div> </a> <ul id="toc-Total_variation_norm_of_vector-valued_measures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_variation_of_probability_measures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Total_variation_of_probability_measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.5</span> <span>Total variation of probability measures</span> </div> </a> <ul id="toc-Total_variation_of_probability_measures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Basic_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Basic properties</span> </div> </a> <button aria-controls="toc-Basic_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Basic properties subsection</span> </button> <ul id="toc-Basic_properties-sublist" class="vector-toc-list"> <li id="toc-Total_variation_of_differentiable_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_variation_of_differentiable_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Total variation of differentiable functions</span> </div> </a> <ul id="toc-Total_variation_of_differentiable_functions-sublist" class="vector-toc-list"> <li id="toc-The_form_of_the_total_variation_of_a_differentiable_function_of_one_variable" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_form_of_the_total_variation_of_a_differentiable_function_of_one_variable"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>The form of the total variation of a differentiable function of one variable</span> </div> </a> <ul id="toc-The_form_of_the_total_variation_of_a_differentiable_function_of_one_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_form_of_the_total_variation_of_a_differentiable_function_of_several_variables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_form_of_the_total_variation_of_a_differentiable_function_of_several_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>The form of the total variation of a differentiable function of several variables</span> </div> </a> <ul id="toc-The_form_of_the_total_variation_of_a_differentiable_function_of_several_variables-sublist" class="vector-toc-list"> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.1</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lemma" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Lemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.2</span> <span>Lemma</span> </div> </a> <ul id="toc-Lemma-sublist" class="vector-toc-list"> <li id="toc-Proof_of_the_lemma" class="vector-toc-list-item vector-toc-level-5"> <a class="vector-toc-link" href="#Proof_of_the_lemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.2.1</span> <span>Proof of the lemma</span> </div> </a> <ul id="toc-Proof_of_the_lemma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_of_the_equality" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Proof_of_the_equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.3</span> <span>Proof of the equality</span> </div> </a> <ul id="toc-Proof_of_the_equality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Total_variation_of_a_measure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_variation_of_a_measure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Total variation of a measure</span> </div> </a> <ul id="toc-Total_variation_of_a_measure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historical_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Historical references</span> </div> </a> <ul id="toc-Historical_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Applications_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Applications</span> </div> </a> <ul id="toc-Applications_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Total variation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 10 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-10" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">10 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Manhikatika" title="Manhikatika – Shona" lang="sn" hreflang="sn" data-title="Manhikatika" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Variation_(Mathematik)" title="Variation (Mathematik) – German" lang="de" hreflang="de" data-title="Variation (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Variation_totale_d%27une_fonction" title="Variation totale d'une fonction – French" lang="fr" hreflang="fr" data-title="Variation totale d'une fonction" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EB%B3%80%EB%8F%99" title="전변동 – Korean" lang="ko" hreflang="ko" data-title="전변동" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Totale_variatie" title="Totale variatie – Dutch" lang="nl" hreflang="nl" data-title="Totale variatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wahanie_funkcji" title="Wahanie funkcji – Polish" lang="pl" hreflang="pl" data-title="Wahanie funkcji" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Varia%C3%A7%C3%A3o_total" title="Variação total – Portuguese" lang="pt" hreflang="pt" data-title="Variação total" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D0%B8%D0%B0%D1%86%D0%B8%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Вариация функции – Russian" lang="ru" hreflang="ru" data-title="Вариация функции" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B0%D1%80%D1%96%D0%B0%D1%86%D1%96%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97" title="Варіація функції – Ukrainian" lang="uk" hreflang="uk" data-title="Варіація функції" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%80%BB%E5%8F%98%E5%B7%AE" title="总变差 – Chinese" lang="zh" hreflang="zh" data-title="总变差" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1936288#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Total_variation" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Total_variation" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Total_variation"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Total_variation&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Total_variation&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Total_variation"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Total_variation&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Total_variation&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Total_variation" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Total_variation" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Total_variation&oldid=1268508491" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Total_variation&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Total_variation&id=1268508491&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FTotal_variation"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FTotal_variation"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Total_variation&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Total_variation&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1936288" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Measure of local oscillation behavior</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Total_variation_distance_of_probability_measures" title="Total variation distance of probability measures">Total variation distance of probability measures</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Primary_sources plainlinks metadata ambox ambox-content ambox-Primary_sources" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies excessively on <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">references</a> to <a href="/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources" title="Wikipedia:No original research">primary sources</a></b>.<span class="hide-when-compact"> Please improve this article by adding <a href="/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources" title="Wikipedia:No original research">secondary or tertiary sources</a>. <br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Total+variation%22">"Total variation"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Total+variation%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Total+variation%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Total+variation%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Total+variation%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Total+variation%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>total variation</b> identifies several slightly different concepts, related to the (<a href="/wiki/Local_property" title="Local property">local</a> or global) structure of the <a href="/wiki/Codomain" title="Codomain">codomain</a> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> or a <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>. For a <a href="/wiki/Real_number" title="Real number">real-valued</a> <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> <i>f</i>, defined on an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> [<i>a</i>, <i>b</i>] ⊂ <b>R</b>, its total variation on the interval of definition is a measure of the one-dimensional <a href="/wiki/Arclength" class="mw-redirect" title="Arclength">arclength</a> of the curve with parametric equation <i>x</i> ↦ <i>f</i>(<i>x</i>), for <i>x</i> ∈ [<i>a</i>, <i>b</i>]. Functions whose total variation is finite are called <i><a href="/wiki/Bounded_variation" title="Bounded variation">functions of bounded variation</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_note">Historical note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=1" title="Edit section: Historical note"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of total variation for functions of one real variable was first introduced by <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a> in the paper (<a href="#CITEREFJordan1881">Jordan 1881</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> He used the new concept in order to prove a convergence theorem for <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> of <a href="/wiki/Discontinuous_function" class="mw-redirect" title="Discontinuous function">discontinuous</a> <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a> whose variation is <a href="/wiki/Bounded_variation" title="Bounded variation">bounded</a>. The extension of the concept to functions of more than one variable however is not simple for various reasons. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=2" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Total_variation_for_functions_of_one_real_variable">Total variation for functions of one real variable</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=3" title="Edit section: Total variation for functions of one real variable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">Definition 1.1.</span> The <b>total variation</b> of a <a href="/wiki/Real_number" title="Real number">real</a>-valued (or more generally <a href="/wiki/Complex_number" title="Complex number">complex</a>-valued) <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, defined on an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>⊂<!-- ⊂ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a659536067aaaac2db1c44613a09a715f0cf7246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.332ex; height:2.843ex;" alt="{\displaystyle [a,b]\subset \mathbb {R} }"></span> is the quantity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{a}^{b}(f)=\sup _{\mathcal {P}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </munder> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{a}^{b}(f)=\sup _{\mathcal {P}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ddc885639da31f7ddc62ea019f0e7e51ba9b78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.885ex; height:7.343ex;" alt="{\displaystyle V_{a}^{b}(f)=\sup _{\mathcal {P}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,}"></span></dd></dl> <p>where the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> runs over the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all <a href="/wiki/Partition_of_an_interval" title="Partition of an interval">partitions</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>∣<!-- ∣ --></mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a partition of </mtext> </mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92752998109b83c4e369a57e35b3a5a420382a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.238ex; height:3.009ex;" alt="{\displaystyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]\right\}}"></span> of the given <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>. Which means that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=x_{0}<x_{1}<...<x_{n_{P}}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=x_{0}<x_{1}<...<x_{n_{P}}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7a30ae4c2bc2c9796a43c18464bdfaf1ac0f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.013ex; height:2.843ex;" alt="{\displaystyle a=x_{0}<x_{1}<...<x_{n_{P}}=b}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Total_variation_for_functions_of_n_>_1_real_variables"><span id="Total_variation_for_functions_of_n_.3E_1_real_variables"></span>Total variation for functions of <i>n</i> > 1 real variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=4" title="Edit section: Total variation for functions of n > 1 real variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Total_variation" title="Special:EditPage/Total variation">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">September 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">Definition 1.2.</span><sup id="cite_ref-10.1093/oso/9780198502456.001.0001_2-0" class="reference"><a href="#cite_note-10.1093/oso/9780198502456.001.0001-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Let <b>Ω</b> be an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of <b>R</b><sup><i>n</i></sup>. Given a function <i>f</i> belonging to <i>L</i><sup>1</sup>(<b>Ω</b>), the <b>total variation</b> of <i>f</i> in <b>Ω</b> is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f(x)\operatorname {div} \phi (x)\,\mathrm {d} x\colon \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \phi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>div</mi> <mo>⁡<!-- --></mo> <mi>ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>:<!-- : --></mo> <mi>ϕ<!-- ϕ --></mi> <mo>∈<!-- ∈ --></mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>ϕ<!-- ϕ --></mi> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>≤<!-- ≤ --></mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f(x)\operatorname {div} \phi (x)\,\mathrm {d} x\colon \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \phi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73cfb84d38f50972a341248ba24c926d27d4eac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:69.163ex; height:6.176ex;" alt="{\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f(x)\operatorname {div} \phi (x)\,\mathrm {d} x\colon \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \phi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},}"></span></dd></dl> <p>where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4befc295df68b89922371dca37fa705a451ec802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.27ex; height:3.009ex;" alt="{\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}"></span> is the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">continuously differentiable</a> <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector functions</a> of <a href="/wiki/Support_(mathematics)#Compact_support" title="Support (mathematics)">compact support</a> contained in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mspace width="thickmathspace" /> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09db6d35cca1004dc2fe6d9e34680fe95246a1cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.286ex; height:3.176ex;" alt="{\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}}"></span> is the <a href="/wiki/Essential_supremum" class="mw-redirect" title="Essential supremum">essential supremum</a> <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982dd33ed8ab34888860f2f24827fa15cd52e1eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.167ex; height:2.176ex;" alt="{\displaystyle \operatorname {div} }"></span> is the <a href="/wiki/Divergence" title="Divergence">divergence</a> operator.</li></ul> <p>This definition <i>does not require</i> that the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo>⊆<!-- ⊆ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2776d402b8e3ef34e3505d645b264f6dfb6a72e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.673ex; height:2.509ex;" alt="{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}"></span> of the given function be a <a href="/wiki/Bounded_set" title="Bounded set">bounded set</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Total_variation_in_measure_theory">Total variation in measure theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=5" title="Edit section: Total variation in measure theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Classical_total_variation_definition">Classical total variation definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=6" title="Edit section: Classical total variation definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Following <a href="#CITEREFSaks1937">Saks (1937</a>, p. 10), consider a <a href="/wiki/Signed_measure" title="Signed measure">signed measure</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> on a <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">measurable space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c4fcf15808c3dbf52cc4b3fd473e97e231f53f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (X,\Sigma )}"></span>: then it is possible to define two <a href="/wiki/Set_function" title="Set function">set functions</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathrm {W} }}(\mu ,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathrm {W} }}(\mu ,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af53aa3062661f3822db14b999caccd115770af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.396ex; height:3.509ex;" alt="{\displaystyle {\overline {\mathrm {W} }}(\mu ,\cdot )}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathrm {W} }}(\mu ,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathrm {W} }}(\mu ,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e9aa4d50dcc9cb1d3332f958fa6aeda3cafe70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:7.283ex; height:3.343ex;" alt="{\displaystyle {\underline {\mathrm {W} }}(\mu ,\cdot )}"></span>, respectively called <b>upper variation</b> and <b>lower variation</b>, as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∣<!-- ∣ --></mo> <mi>A</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>A</mi> <mo>⊂<!-- ⊂ --></mo> <mi>E</mi> </mrow> <mo>}</mo> </mrow> <mspace width="2em" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>E</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a158a33add91118455ed05e52ab86d4ed8bc52a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.134ex; height:3.509ex;" alt="{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mrow> <mo>{</mo> <mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∣<!-- ∣ --></mo> <mi>A</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>A</mi> <mo>⊂<!-- ⊂ --></mo> <mi>E</mi> </mrow> <mo>}</mo> </mrow> <mspace width="2em" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>E</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/468e146ad85d8ee5e8ddc0a338b79f40af2896dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:54.172ex; height:3.343ex;" alt="{\displaystyle {\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }"></span></dd></dl> <p>clearly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>≥<!-- ≥ --></mo> <mn>0</mn> <mo>≥<!-- ≥ --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>E</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b16a5adc92d943c7f75d17f9a00b06953f42d787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:36.527ex; height:4.009ex;" alt="{\displaystyle {\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma }"></span></dd></dl> <p><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">Definition 1.3.</span> The <b>variation</b> (also called <b>absolute variation</b>) of the signed measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is the set function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="2em" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>E</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461c78003295d8535f78e179aaf8b32048f85bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:43.068ex; height:4.009ex;" alt="{\displaystyle |\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma }"></span></dd></dl> <p>and its <b>total variation</b> is defined as the value of this measure on the whole space of definition, i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mu \|=|\mu |(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>μ<!-- μ --></mi> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mu \|=|\mu |(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6063907fe3331b421c8e2ea4dd53bad9d5db4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.31ex; height:2.843ex;" alt="{\displaystyle \|\mu \|=|\mu |(X)}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Modern_definition_of_total_variation_norm">Modern definition of total variation norm</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=7" title="Edit section: Modern definition of total variation norm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFSaks1937">Saks (1937</a>, p. 11) uses upper and lower variations to prove the <a href="/wiki/Hahn_decomposition_theorem" title="Hahn decomposition theorem">Hahn–Jordan decomposition</a>: according to his version of this theorem, the upper and lower variation are respectively a <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> and a <a href="/wiki/Non-positive" class="mw-redirect" title="Non-positive">non-positive</a> <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>. Using a more modern notation, define </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9fd344fd436bd78ae301549acee5c77b9198e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.897ex; height:3.509ex;" alt="{\displaystyle \mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> <mo stretchy="false">(</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e803b7070a5918fda8c86fcafd27acad7ec4fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:18.592ex; height:3.509ex;" alt="{\displaystyle \mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,}"></span></dd></dl> <p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efba17f582e5e2330f9c3166ea6ae8b4b53e0ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.912ex; height:3.009ex;" alt="{\displaystyle \mu ^{+}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1052ef22be7feef21f5a196d2a08d30d3e9835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.912ex; height:3.009ex;" alt="{\displaystyle \mu ^{-}}"></span> are two non-negative <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measures</a> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu ^{+}-\mu ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> <mo>=</mo> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu ^{+}-\mu ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94507e88b00801e0f0d1a891f933462b2a393678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.166ex; height:3.009ex;" alt="{\displaystyle \mu =\mu ^{+}-\mu ^{-}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5656b81ac6a464580ddeb0ff0085408b5320b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.459ex; height:3.009ex;" alt="{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}"></span></dd></dl> <p>The last measure is sometimes called, by <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>, <b>total variation measure</b>. </p> <div class="mw-heading mw-heading4"><h4 id="Total_variation_norm_of_complex_measures">Total variation norm of complex measures</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=8" title="Edit section: Total variation norm of complex measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is <a href="/wiki/Complex_number" title="Complex number">complex-valued</a> i.e. is a <a href="/wiki/Complex_measure" title="Complex measure">complex measure</a>, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow <a href="#CITEREFRudin1966">Rudin (1966</a>, pp. 137–139) and define the total variation of the complex-valued measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> as follows </p><p><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">Definition 1.4.</span> The <b>variation</b> of the complex-valued measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is the <a href="/wiki/Set_function" title="Set function">set function</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π<!-- π --></mi> </mrow> </munder> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∈<!-- ∈ --></mo> <mi>π<!-- π --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="2em" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>E</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e849c75cd5a2d56b6678534553a2450e22855d4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.489ex; height:5.676ex;" alt="{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma }"></span></dd></dl> <p>where the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> is taken over all partitions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> of a <a href="/wiki/Measurable_set" class="mw-redirect" title="Measurable set">measurable set</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> into a countable number of disjoint measurable subsets. </p><p>This definition coincides with the above definition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5656b81ac6a464580ddeb0ff0085408b5320b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.459ex; height:3.009ex;" alt="{\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}"></span> for the case of real-valued signed measures. </p> <div class="mw-heading mw-heading4"><h4 id="Total_variation_norm_of_vector-valued_measures">Total variation norm of vector-valued measures</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=9" title="Edit section: Total variation norm of vector-valued measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The variation so defined is a <a href="/wiki/Positive_measure" class="mw-redirect" title="Positive measure">positive measure</a> (see <a href="#CITEREFRudin1966">Rudin (1966</a>, p. 139)) and coincides with the one defined by <b><a href="#math_3">1.3</a></b> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is a <a href="/wiki/Signed_measure" title="Signed measure">signed measure</a>: its total variation is defined as above. This definition works also if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is a <a href="/wiki/Vector_measure" title="Vector measure">vector measure</a>: the variation is then defined by the following formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }\|\mu (A)\|\qquad \forall E\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π<!-- π --></mi> </mrow> </munder> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∈<!-- ∈ --></mo> <mi>π<!-- π --></mi> </mrow> </munder> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mspace width="2em" /> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>E</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }\|\mu (A)\|\qquad \forall E\in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2fdbb72f44c9530eba990e7b699a158ab10b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.52ex; height:5.676ex;" alt="{\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }\|\mu (A)\|\qquad \forall E\in \Sigma }"></span></dd></dl> <p>where the supremum is as above. This definition is slightly more general than the one given by <a href="#CITEREFRudin1966">Rudin (1966</a>, p. 138) since it requires only to consider <i>finite partitions</i> of the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>: this implies that it can be used also to define the total variation on <a href="/wiki/Sigma_additivity" class="mw-redirect" title="Sigma additivity">finite-additive measures</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Total_variation_of_probability_measures">Total variation of probability measures</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=10" title="Edit section: Total variation of probability measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Total_variation" title="Special:EditPage/Total variation">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Total_variation_distance_of_probability_measures" title="Total variation distance of probability measures">Total variation distance of probability measures</a></div> <p>The total variation of any <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are <a href="/wiki/Probability_measure" title="Probability measure">probability measures</a>, the <b><a href="/wiki/Total_variation_distance_of_probability_measures" title="Total variation distance of probability measures">total variation distance of probability measures</a></b> can be defined as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mu -\nu \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>μ<!-- μ --></mi> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mu -\nu \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860d1d09b03680393673f56e4c1b90b18720323d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.799ex; height:2.843ex;" alt="{\displaystyle \|\mu -\nu \|}"></span> where the norm is the total variation norm of signed measures. Using the property that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu -\nu )(X)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu -\nu )(X)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849f02664205d9b4966dab4f9a88199dc4ec49ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.334ex; height:2.843ex;" alt="{\displaystyle (\mu -\nu )(X)=0}"></span>, we eventually arrive at the equivalent definition </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>μ<!-- μ --></mi> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>μ<!-- μ --></mi> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>:</mo> <mi>A</mi> <mo>∈<!-- ∈ --></mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mspace width="thinmathspace" /> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc01a6a5edf7326174a190be6a896ed98cc9e65f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.162ex; height:2.843ex;" alt="{\displaystyle \|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}}"></span></dd></dl> <p>and its values are non-trivial. The factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> above is usually dropped (as is the convention in the article <a href="/wiki/Total_variation_distance_of_probability_measures" title="Total variation distance of probability measures">total variation distance of probability measures</a>). Informally, this is the largest possible difference between the probabilities that the two <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> can assign to the same event. For a <a href="/wiki/Categorical_distribution" title="Categorical distribution">categorical distribution</a> it is possible to write the total variation distance as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb122e588c2235467def9b81e9c87a8075e5ef0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.091ex; height:5.509ex;" alt="{\displaystyle \delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.}"></span></dd></dl> <p>It may also be normalized to values in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> by halving the previous definition as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f9d620196e31879ce24c3aa1e2a4f5124e6fc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.798ex; height:6.343ex;" alt="{\displaystyle \delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|}"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=11" title="Edit section: Basic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Total_variation_of_differentiable_functions">Total variation of differentiable functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=12" title="Edit section: Total variation of differentiable functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The total variation of a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}({\overline {\Omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}({\overline {\Omega }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2bae12b835ef0b9b93f8416213a372eaf1cabf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.454ex; height:3.509ex;" alt="{\displaystyle C^{1}({\overline {\Omega }})}"></span> function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> can be expressed as an <a href="/wiki/Integral" title="Integral">integral</a> involving the given function instead of as the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> of the <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functionals</a> of definitions <b><a href="#math_1">1.1</a></b> and <b><a href="#math_2">1.2</a></b>. </p> <div class="mw-heading mw-heading4"><h4 id="The_form_of_the_total_variation_of_a_differentiable_function_of_one_variable">The form of the total variation of a differentiable function of one variable</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=13" title="Edit section: The form of the total variation of a differentiable function of one variable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">Theorem 1.</span> The <b>total variation</b> of a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, defined on an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>⊂<!-- ⊂ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a659536067aaaac2db1c44613a09a715f0cf7246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.332ex; height:2.843ex;" alt="{\displaystyle [a,b]\subset \mathbb {R} }"></span>, has the following expression if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> is Riemann integrable </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{a}^{b}(f)=\int _{a}^{b}|f'(x)|\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{a}^{b}(f)=\int _{a}^{b}|f'(x)|\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd3bbf97e41df1cf47fec7d9bb917d946fe2258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.89ex; height:6.343ex;" alt="{\displaystyle V_{a}^{b}(f)=\int _{a}^{b}|f'(x)|\mathrm {d} x}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is differentiable and <a href="/wiki/Monotonic_function" title="Monotonic function">monotonic</a>, then the above simplifies to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{a}^{b}(f)=|f(a)-f(b)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{a}^{b}(f)=|f(a)-f(b)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33b550f10b69a0bab8981bb3a9469e57387feb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.578ex; height:3.009ex;" alt="{\displaystyle V_{a}^{b}(f)=|f(a)-f(b)|}"></span></dd></dl> <p>For any differentiable function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, we can decompose the domain interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>, into subintervals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,a_{1}],[a_{1},a_{2}],\dots ,[a_{N},b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,a_{1}],[a_{1},a_{2}],\dots ,[a_{N},b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42923d76f397481bf7aa71b391ce205426c1cc50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.196ex; height:2.843ex;" alt="{\displaystyle [a,a_{1}],[a_{1},a_{2}],\dots ,[a_{N},b]}"></span> (with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<a_{1}<a_{2}<\cdots <a_{N}<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mo>⋯<!-- ⋯ --></mo> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<a_{1}<a_{2}<\cdots <a_{N}<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7524cd29456318b15c589f5e40510fb10fd0f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.932ex; height:2.509ex;" alt="{\displaystyle a<a_{1}<a_{2}<\cdots <a_{N}<b}"></span>) in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is locally monotonic, then the total variation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> can be written as the sum of local variations on those subintervals: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\[0.3em]&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋯<!-- ⋯ --></mo> <mspace width="thinmathspace" /> <mo>+</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋯<!-- ⋯ --></mo> <mspace width="thinmathspace" /> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\[0.3em]&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd29e6a29f9d85a967feabb7d366f6428e7b2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.741ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\[0.3em]&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="The_form_of_the_total_variation_of_a_differentiable_function_of_several_variables">The form of the total variation of a differentiable function of several variables</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=14" title="Edit section: The form of the total variation of a differentiable function of several variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">Theorem 2.</span> Given a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}({\overline {\Omega }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}({\overline {\Omega }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2bae12b835ef0b9b93f8416213a372eaf1cabf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.454ex; height:3.509ex;" alt="{\displaystyle C^{1}({\overline {\Omega }})}"></span> function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> defined on a <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> <a href="/wiki/Open_set" title="Open set">open set</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo>⊆<!-- ⊆ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2776d402b8e3ef34e3505d645b264f6dfb6a72e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.673ex; height:2.509ex;" alt="{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16feddaad462c2a1c9efdaeee062a0484a023fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }"></span> of class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}"></span>, the <b>total variation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></b> has the following expression </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(f,\Omega )=\int _{\Omega }\left|\nabla f(x)\right|\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi mathvariant="normal">Ω<!-- Ω --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(f,\Omega )=\int _{\Omega }\left|\nabla f(x)\right|\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d9322edde1878f98c46087467e95f2d354a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.44ex; height:5.676ex;" alt="{\displaystyle V(f,\Omega )=\int _{\Omega }\left|\nabla f(x)\right|\mathrm {d} x}"></span> .</dd></dl> <div class="mw-heading mw-heading5"><h5 id="Proof">Proof</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=15" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first step in the proof is to first prove an equality which follows from the <a href="/wiki/Gauss%E2%80%93Ostrogradsky_theorem" class="mw-redirect" title="Gauss–Ostrogradsky theorem">Gauss–Ostrogradsky theorem</a>. </p> <div class="mw-heading mw-heading5"><h5 id="Lemma">Lemma</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=16" title="Edit section: Lemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under the conditions of the theorem, the following equality holds: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mi>div</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mo>=</mo> <mo>−<!-- − --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>⋅<!-- ⋅ --></mo> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97be96a379705846b2a9704c2925a5c498daa024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.644ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }"></span></dd></dl> <div class="mw-heading mw-heading6"><h6 id="Proof_of_the_lemma">Proof of the lemma</h6><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=17" title="Edit section: Proof of the lemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the <a href="/wiki/Gauss%E2%80%93Ostrogradsky_theorem" class="mw-redirect" title="Gauss–Ostrogradsky theorem">Gauss–Ostrogradsky theorem</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f9b66bfe44425c91e71f0498a414b83f870012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.952ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} }"></span></dd></dl> <p>by substituting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} :=f\mathbf {\varphi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>:=</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} :=f\mathbf {\varphi } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/746f01edd5536c24d0fea3d9fe497d38bcb6c8c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.547ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} :=f\mathbf {\varphi } }"></span>, we have: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=\int _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=\int _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d22ffd293b6f45dec4dc679efafb80581d2975e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.775ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=\int _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} }"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\varphi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\varphi } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a872328945f486a049f15dd5203723b8841767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \mathbf {\varphi } }"></span> is zero on the border of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> by definition: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3387fc005e3a19ed20a65d0a3e26c7f83cbe0600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.134ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4ada40753c7e4dcef8abd2016dcc13002c6ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.186ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>f</mi> <mo>+</mo> <mi>f</mi> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/994a3d1fc50b04b2e60e09f4658b0fb752d67cd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.46ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d660e87724041b7f92dd202f769a23a937f18d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.751ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>=</mo> <mo>−<!-- − --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46add30d8a561d303c51716ad2f1dc274dc90458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.644ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Proof_of_the_equality">Proof of the equality</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=18" title="Edit section: Proof of the equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under the conditions of the theorem, from the lemma we have: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>=</mo> <mo>−<!-- − --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>≤<!-- ≤ --></mo> <mrow> <mo>|</mo> <mrow> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> <mo>≤<!-- ≤ --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>|</mo> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> <mo>≤<!-- ≤ --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a18f47ea7da6fc5deea855d6ef25dedee82564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:64.451ex; height:5.843ex;" alt="{\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|}"></span></dd></dl> <p>in the last part <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\varphi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\varphi } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a872328945f486a049f15dd5203723b8841767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \mathbf {\varphi } }"></span> could be omitted, because by definition its essential supremum is at most one. </p><p>On the other hand, we consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{N}:=-\mathbb {I} _{\left[-N,N\right]}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ<!-- θ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>:=</mo> <mo>−<!-- − --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mo>−<!-- − --></mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{N}:=-\mathbb {I} _{\left[-N,N\right]}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d95bf9d95132230ccaf3af9196b1c45f28fa00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.54ex; height:6.176ex;" alt="{\displaystyle \theta _{N}:=-\mathbb {I} _{\left[-N,N\right]}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{N}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>θ<!-- θ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{N}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be4738364acf437b9dddcaaa3aa79427bb21b344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.782ex; height:2.843ex;" alt="{\displaystyle \theta _{N}^{*}}"></span> which is the up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε<!-- ε --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ<!-- θ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fa0430c35c017ef5f7404284a8481e9af97d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.782ex; height:2.509ex;" alt="{\displaystyle \theta _{N}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{c}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{c}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a547bd72b788eaa51bec8d85ae1007ed261dce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{c}^{1}}"></span> with the same integral. We can do this since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{c}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{c}^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a547bd72b788eaa51bec8d85ae1007ed261dce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{c}^{1}}"></span> is dense in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"></span>. Now again substituting into the lemma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\[4pt]&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left[-N,N\right]}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\lim _{N\to \infty }\int _{\left[-N,N\right]\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mi>div</mi> <mo>⁡<!-- --></mo> <msubsup> <mi>θ<!-- θ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mo>−<!-- − --></mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> </mrow> </msub> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mo>−<!-- − --></mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> </mrow> <mo>]</mo> </mrow> <mo>∩<!-- ∩ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </mrow> </msub> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\[4pt]&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left[-N,N\right]}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\lim _{N\to \infty }\int _{\left[-N,N\right]\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463f3c5b1ed80954483b0b794762ded3990c36bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.171ex; width:33.856ex; height:27.509ex;" alt="{\displaystyle {\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\[4pt]&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left[-N,N\right]}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\lim _{N\to \infty }\int _{\left[-N,N\right]\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}}"></span></dd></dl> <p>This means we have a convergent sequence of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edc7eba6a5585f9216b4802c58e587983b3c6c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.643ex; height:3.176ex;" alt="{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } }"></span> that tends to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{\Omega }\left|\nabla f\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{\Omega }\left|\nabla f\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d1873534d0cb18e7fd1fdc67e0a372e6ba94b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.412ex; height:3.176ex;" alt="{\textstyle \int _{\Omega }\left|\nabla f\right|}"></span> as well as we know that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mi>f</mi> <mi>div</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> <mo>≤<!-- ≤ --></mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01fcc1299f78c5448032fc109c157d2f24a256c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.153ex; height:3.176ex;" alt="{\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|}"></span>. <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> </p><p>It can be seen from the proof that the supremum is attained when </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>f</mi> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e149ca306c4186b2bbb36eff716f02840e3fe821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.64ex; height:6.176ex;" alt="{\displaystyle \varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.}"></span></dd></dl> <p>The <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is said to be of <a href="/wiki/Bounded_variation" title="Bounded variation">bounded variation</a> precisely if its total variation is finite. </p> <div class="mw-heading mw-heading3"><h3 id="Total_variation_of_a_measure">Total variation of a measure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=19" title="Edit section: Total variation of a measure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The total variation is a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, called the <a href="/wiki/Ca_space" class="mw-redirect" title="Ca space">ca space</a>, relative to this norm. It is contained in the larger Banach space, called the <a href="/wiki/Ba_space" title="Ba space">ba space</a>, consisting of <i><a href="/wiki/Finitely_additive_measure" class="mw-redirect" title="Finitely additive measure">finitely additive</a></i> (as opposed to countably additive) measures, also with the same norm. The <a href="/wiki/Distance_function" class="mw-redirect" title="Distance function">distance function</a> associated to the norm gives rise to the total variation distance between two measures <i>μ</i> and <i>ν</i>. </p><p>For finite measures on <b>R</b>, the link between the total variation of a measure <i>μ</i> and the total variation of a function, as described above, goes as follows. Given <i>μ</i>, define a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>:<!-- : --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b81b16317269d7796f28bf9c4bc3ac228095deb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.524ex; height:2.676ex;" alt="{\displaystyle \varphi \colon \mathbb {R} \to \mathbb {R} }"></span> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)=\mu ((-\infty ,t])~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)=\mu ((-\infty ,t])~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c351e788d819177544477304f92d385ad49fe788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.263ex; height:2.843ex;" alt="{\displaystyle \varphi (t)=\mu ((-\infty ,t])~.}"></span></dd></dl> <p>Then, the total variation of the signed measure <i>μ</i> is equal to the total variation, in the above sense, of the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>. In general, the total variation of a signed measure can be defined using <a href="/wiki/Hahn_decomposition_theorem" title="Hahn decomposition theorem">Jordan's decomposition theorem</a> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mi>μ<!-- μ --></mi> <msub> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00cfa7dfc76dc95d8ac5ee3036672935ab7a2bbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.95ex; height:2.843ex;" alt="{\displaystyle \|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,}"></span></dd></dl> <p>for any signed measure <i>μ</i> on a measurable space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c4fcf15808c3dbf52cc4b3fd473e97e231f53f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (X,\Sigma )}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=20" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Total variation can be seen as a <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> <a href="/wiki/Real_number" title="Real number">real</a>-valued <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a> defined on the space of <a href="/wiki/Real_number" title="Real number">real-valued</a> <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> (for the case of functions of one variable) or on the space of <a href="/wiki/Integrable_function" class="mw-redirect" title="Integrable function">integrable functions</a> (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like <a href="/wiki/Optimal_control" title="Optimal control">optimal control</a>, <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, and <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>, where the solution to a certain problem has to <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">minimize</a> its value. As an example, use of the total variation functional is common in the following two kind of problems </p> <ul><li><b>Numerical analysis of differential equations</b>: it is the science of finding approximate solutions to <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. Applications of total variation to these problems are detailed in the article "<i><a href="/wiki/Total_variation_diminishing" title="Total variation diminishing">total variation diminishing</a></i>"</li> <li><b>Image denoising</b>:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> in <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, denoising is a collection of methods used to reduce the <a href="/wiki/Electronic_noise" class="mw-redirect" title="Electronic noise">noise</a> in an <a href="/wiki/Image" title="Image">image</a> reconstructed from data obtained by electronic means, for example <a href="/wiki/Data_transmission" class="mw-redirect" title="Data transmission">data transmission</a> or <a href="/wiki/Sensor" title="Sensor">sensing</a>. "<i><a href="/wiki/Total_variation_denoising" title="Total variation denoising">Total variation denoising</a></i>" is the name for the application of total variation to image noise reduction; further details can be found in the papers of (<a href="#CITEREFRudinOsherFatemi1992">Rudin, Osher & Fatemi 1992</a>) and (<a href="#CITEREFCasellesChambolleNovaga2007">Caselles, Chambolle & Novaga 2007</a>). A sensible extension of this model to colour images, called Colour TV, can be found in (<a href="#CITEREFBlomgrenChan1998">Blomgren & Chan 1998</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bounded_variation" title="Bounded variation">Bounded variation</a></li> <li><a href="/wiki/P-variation" title="P-variation">p-variation</a></li> <li><a href="/wiki/Total_variation_diminishing" title="Total variation diminishing">Total variation diminishing</a></li> <li><a href="/wiki/Total_variation_denoising" title="Total variation denoising">Total variation denoising</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Total_variation_distance_of_probability_measures" title="Total variation distance of probability measures">Total variation distance of probability measures</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov test</a></li> <li><a href="/wiki/Anisotropic_diffusion" title="Anisotropic diffusion">Anisotropic diffusion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=22" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">February 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">According to <a href="#CITEREFGolubovVitushkin2001">Golubov & Vitushkin (2001)</a>.</span> </li> <li id="cite_note-10.1093/oso/9780198502456.001.0001-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-10.1093/oso/9780198502456.001.0001_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAmbrosioFuscoPallara2000" class="citation book cs1">Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1093/oso/9780198502456.001.0001}"><i>Functions of Bounded Variation and Free Discontinuity Problems</i></a>. Oxford University Press. p. 119. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Foso%2F9780198502456.001.0001">10.1093/oso/9780198502456.001.0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198502456" title="Special:BookSources/9780198502456"><bdi>9780198502456</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+Bounded+Variation+and+Free+Discontinuity+Problems&rft.pages=119&rft.pub=Oxford+University+Press&rft.date=2000&rft_id=info%3Adoi%2F10.1093%2Foso%2F9780198502456.001.0001&rft.isbn=9780198502456&rft.aulast=Ambrosio&rft.aufirst=Luigi&rft.au=Fusco%2C+Nicola&rft.au=Pallara%2C+Diego&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%2Foso%2F9780198502456.001.0001%7D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbsFrancis_Edward_Su2002" class="citation web cs1">Gibbs, Alison; Francis Edward Su (2002). <a rel="nofollow" class="external text" href="https://www.math.hmc.edu/~su/papers.dir/metrics.pdf">"On Choosing and Bounding Probability Metrics"</a> <span class="cs1-format">(PDF)</span>. p. 7<span class="reference-accessdate">. Retrieved <span class="nowrap">8 April</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=On+Choosing+and+Bounding+Probability+Metrics&rft.pages=7&rft.date=2002&rft.aulast=Gibbs&rft.aufirst=Alison&rft.au=Francis+Edward+Su&rft_id=https%3A%2F%2Fwww.math.hmc.edu%2F~su%2Fpapers.dir%2Fmetrics.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://arxiv.org/pdf/1603.09599">https://arxiv.org/pdf/1603.09599</a> Retrieved 12/15/2024</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Historical_references">Historical references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=23" title="Edit section: Historical references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArzelà1905" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Cesare_Arzel%C3%A0" title="Cesare Arzelà">Arzelà, Cesare</a> (7 May 1905), <a rel="nofollow" class="external text" href="https://archive.org/stream/rendicontodelle04bologoog#page/n121/mode/2up">"Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)"</a>, <i>Rendiconto delle Sessioni della Reale Accademia delle Scienze dell'Istituto di Bologna</i>, Nuova serie (in Italian), <b>IX</b> (4): <span class="nowrap">100–</span>107, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:36.0491.02">36.0491.02</a>, archived from <a rel="nofollow" class="external text" href="https://archive.org/details/rendicontodelle04bologoog">the original</a> on 2007-08-07</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconto+delle+Sessioni+della+Reale+Accademia+delle+Scienze+dell%27Istituto+di+Bologna&rft.atitle=Sulle+funzioni+di+due+variabili+a+variazione+limitata+%28On+functions+of+two+variables+of+bounded+variation%29&rft.volume=IX&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E100-%3C%2Fspan%3E107&rft.date=1905-05-07&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A36.0491.02%23id-name%3DJFM&rft.aulast=Arzel%C3%A0&rft.aufirst=Cesare&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frendicontodelle04bologoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, Boris I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Arzelà_variation">"Arzelà variation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Arzel%C3%A0+variation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DArzel%C3%A0_variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, Boris I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Fréchet_variation">"Fréchet variation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fr%C3%A9chet+variation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFr%C3%A9chet_variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, Boris I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Hardy_variation">"Hardy variation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hardy+variation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DHardy_variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, Boris I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Pierpont_variation">"Pierpont variation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Pierpont+variation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPierpont_variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, Boris I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Vitali_variation">"Vitali variation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Vitali+variation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DVitali_variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubov2001" class="citation cs2">Golubov, Boris I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Tonelli_plane_variation">"Tonelli plane variation"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tonelli+plane+variation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTonelli_plane_variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubovVitushkin2001" class="citation cs2">Golubov, Boris I.; <a href="/wiki/Anatoli_Georgievich_Vitushkin" class="mw-redirect" title="Anatoli Georgievich Vitushkin">Vitushkin, Anatoli G.</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Variation_of_a_function">"Variation of a function"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Variation+of+a+function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Golubov&rft.aufirst=Boris+I.&rft.au=Vitushkin%2C+Anatoli+G.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DVariation_of_a_function&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJordan1881" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan, Camille</a> (1881), <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227">"Sur la série de Fourier"</a>, <i>Comptes rendus hebdomadaires des séances de l'Académie des sciences</i> (in French), <b>92</b>: <span class="nowrap">228–</span>230, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:13.0184.01">13.0184.01</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+rendus+hebdomadaires+des+s%C3%A9ances+de+l%27Acad%C3%A9mie+des+sciences&rft.atitle=Sur+la+s%C3%A9rie+de+Fourier&rft.volume=92&rft.pages=%3Cspan+class%3D%22nowrap%22%3E228-%3C%2Fspan%3E230&rft.date=1881&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A13.0184.01%23id-name%3DJFM&rft.aulast=Jordan&rft.aufirst=Camille&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k7351t%2Ff227&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span> (available at <a href="/wiki/Gallica" class="mw-redirect" title="Gallica">Gallica</a>). This is, according to Boris Golubov, the first paper on functions of bounded variation.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHahn1921" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Hans_Hahn_(mathematician)" title="Hans Hahn (mathematician)">Hahn, Hans</a> (1921), <a rel="nofollow" class="external text" href="https://archive.org/details/theoriederreelle01hahnuoft"><i>Theorie der reellen Funktionen</i></a> (in German), Berlin: Springer Verlag, pp. VII+600, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:48.0261.09">48.0261.09</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+der+reellen+Funktionen&rft.place=Berlin&rft.pages=VII%2B600&rft.pub=Springer+Verlag&rft.date=1921&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A48.0261.09%23id-name%3DJFM&rft.aulast=Hahn&rft.aufirst=Hans&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoriederreelle01hahnuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVitali1908" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Giuseppe_Vitali" title="Giuseppe Vitali">Vitali, Giuseppe</a> (1908) [17 dicembre 1907], <a rel="nofollow" class="external text" href="https://archive.org/stream/attidellarealeac43real#page/228/mode/2up">"Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)"</a>, <i><a href="/w/index.php?title=Atti_dell%27Accademia_delle_Scienze_di_Torino&action=edit&redlink=1" class="new" title="Atti dell'Accademia delle Scienze di Torino (page does not exist)">Atti dell'Accademia delle Scienze di Torino</a></i> (in Italian), <b>43</b>: <span class="nowrap">75–</span>92, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:39.0101.05">39.0101.05</a>, archived from <a rel="nofollow" class="external text" href="https://archive.org/details/attidellarealeac43real">the original</a> on 2009-03-31</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Atti+dell%27Accademia+delle+Scienze+di+Torino&rft.atitle=Sui+gruppi+di+punti+e+sulle+funzioni+di+variabili+reali+%28On+groups+of+points+and+functions+of+real+variables%29&rft.volume=43&rft.pages=%3Cspan+class%3D%22nowrap%22%3E75-%3C%2Fspan%3E92&rft.date=1908&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A39.0101.05%23id-name%3DJFM&rft.aulast=Vitali&rft.aufirst=Giuseppe&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fattidellarealeac43real&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>. The paper containing the first proof of <a href="/wiki/Vitali_covering_theorem" class="mw-redirect" title="Vitali covering theorem">Vitali covering theorem</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamsClarkson1933" class="citation cs2">Adams, C. Raymond; Clarkson, James A. (1933), "On definitions of bounded variation for functions of two variables", <i><a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>, <b>35</b> (4): <span class="nowrap">824–</span>854, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1933-1501718-2">10.1090/S0002-9947-1933-1501718-2</a></span>, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:59.0285.01">59.0285.01</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1501718">1501718</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0008.00602">0008.00602</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=On+definitions+of+bounded+variation+for+functions+of+two+variables&rft.volume=35&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E824-%3C%2Fspan%3E854&rft.date=1933&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0008.00602%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1501718%23id-name%3DMR&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A59.0285.01%23id-name%3DJFM&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1933-1501718-2&rft.aulast=Adams&rft.aufirst=C.+Raymond&rft.au=Clarkson%2C+James+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCesari1936" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Lamberto_Cesari" title="Lamberto Cesari">Cesari, Lamberto</a> (1936), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=ASNSP_1936_2_5_3-4_299_0">"Sulle funzioni a variazione limitata (On the functions of bounded variation)"</a>, <i><a href="/w/index.php?title=Annali_della_Scuola_Normale_Superiore&action=edit&redlink=1" class="new" title="Annali della Scuola Normale Superiore (page does not exist)">Annali della Scuola Normale Superiore</a></i>, II (in Italian), <b>5</b> (<span class="nowrap">3–</span>4): <span class="nowrap">299–</span>313, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:62.0247.03">62.0247.03</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1556778">1556778</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0014.29605">0014.29605</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annali+della+Scuola+Normale+Superiore&rft.atitle=Sulle+funzioni+a+variazione+limitata+%28On+the+functions+of+bounded+variation%29&rft.volume=5&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E299-%3C%2Fspan%3E313&rft.date=1936&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0014.29605%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1556778%23id-name%3DMR&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A62.0247.03%23id-name%3DJFM&rft.aulast=Cesari&rft.aufirst=Lamberto&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASNSP_1936_2_5_3-4_299_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>. Available at <a rel="nofollow" class="external text" href="http://www.numdam.org">Numdam</a>.</li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeoni2017" class="citation cs2">Leoni, Giovanni (2017), <i>A First Course in Sobolev Spaces: Second Edition</i>, Graduate Studies in Mathematics, American Mathematical Society, pp. xxii+734, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4704-2921-8" title="Special:BookSources/978-1-4704-2921-8"><bdi>978-1-4704-2921-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Sobolev+Spaces%3A+Second+Edition&rft.series=Graduate+Studies+in+Mathematics&rft.pages=xxii%2B734&rft.pub=American+Mathematical+Society&rft.date=2017&rft.isbn=978-1-4704-2921-8&rft.aulast=Leoni&rft.aufirst=Giovanni&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaks1937" class="citation book cs1"><a href="/wiki/Stanis%C5%82aw_Saks" title="Stanisław Saks">Saks, Stanisław</a> (1937). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl"><i>Theory of the Integral</i></a>. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa–Lwów: G.E. Stechert & Co. pp. VI+347. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:63.0183.05">63.0183.05</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0017.30004">0017.30004</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+the+Integral&rft.place=Warszawa%E2%80%93Lw%C3%B3w&rft.series=Monografie+Matematyczne&rft.pages=VI%2B347&rft.edition=2nd&rft.pub=G.E.+Stechert+%26+Co.&rft.date=1937&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0017.30004%23id-name%3DZbl&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A63.0183.05%23id-name%3DJFM&rft.aulast=Saks&rft.aufirst=Stanis%C5%82aw&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fkstresc.php%3Ftom%3D7%26wyd%3D10%26jez%3Dpl&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>. (available at the <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksspis.php?wyd=10&jez=pl">Polish Virtual Library of Science</a>). English translation from the original French by <a href="/wiki/Laurence_Chisholm_Young" title="Laurence Chisholm Young">Laurence Chisholm Young</a>, with two additional notes by <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1966" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1966), <i>Real and Complex Analysis</i>, McGraw-Hill Series in Higher Mathematics (1st ed.), New York: McGraw-Hill, pp. xi+412, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0210528">0210528</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0142.01701">0142.01701</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Complex+Analysis&rft.place=New+York&rft.series=McGraw-Hill+Series+in+Higher+Mathematics&rft.pages=xi%2B412&rft.edition=1st&rft.pub=McGraw-Hill&rft.date=1966&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0142.01701%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D210528%23id-name%3DMR&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>One variable</b> </p> <ul><li>"<a rel="nofollow" class="external text" href="https://planetmath.org/encyclopedia/TotalVariation.html">Total variation</a>" on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li></ul> <p><b>One and more variables</b> </p> <ul><li><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation">Function of bounded variation</a> at <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/">Encyclopedia of Mathematics</a></li></ul> <p><b>Measure theory</b> </p> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Total_Variation"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Rowland, Todd. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TotalVariation.html">"Total Variation"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Total+Variation&rft.au=Rowland%2C+Todd&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTotalVariation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span></span>.</li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/JordanDecomposition">Jordan decomposition</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>..</li> <li><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Jordan_decomposition_%28of_a_signed_measure%29">Jordan decomposition</a> at <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/">Encyclopedia of Mathematics</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Applications_2">Applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Total_variation&action=edit&section=26" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasellesChambolleNovaga2007" class="citation cs2">Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110927172158/http://cvgmt.sns.it/papers/caschanov07/"><i>The discontinuity set of solutions of the TV denoising problem and some extensions</i></a>, <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>, Multiscale Modeling and Simulation, vol. 6 n. 3, archived from <a rel="nofollow" class="external text" href="http://cvgmt.sns.it/papers/caschanov07/">the original</a> on 2011-09-27</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+discontinuity+set+of+solutions+of+the+TV+denoising+problem+and+some+extensions&rft.pub=SIAM%2C+Multiscale+Modeling+and+Simulation%2C+vol.+6+n.+3&rft.date=2007&rft.aulast=Caselles&rft.aufirst=Vicent&rft.au=Chambolle%2C+Antonin&rft.au=Novaga%2C+Matteo&rft_id=http%3A%2F%2Fcvgmt.sns.it%2Fpapers%2Fcaschanov07%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span> (a work dealing with total variation application in denoising problems for <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>).</li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudinOsherFatemi1992" class="citation cs2">Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992), "Nonlinear total variation based noise removal algorithms", <i>Physica D: Nonlinear Phenomena</i>, <b>60</b> (<span class="nowrap">1–</span>4), Physica D: Nonlinear Phenomena 60.1: 259-268: <span class="nowrap">259–</span>268, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992PhyD...60..259R">1992PhyD...60..259R</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-2789%2892%2990242-F">10.1016/0167-2789(92)90242-F</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+D%3A+Nonlinear+Phenomena&rft.atitle=Nonlinear+total+variation+based+noise+removal+algorithms&rft.volume=60&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E259-%3C%2Fspan%3E268&rft.date=1992&rft_id=info%3Adoi%2F10.1016%2F0167-2789%2892%2990242-F&rft_id=info%3Abibcode%2F1992PhyD...60..259R&rft.aulast=Rudin&rft.aufirst=Leonid+I.&rft.au=Osher%2C+Stanley&rft.au=Fatemi%2C+Emad&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlomgrenChan1998" class="citation cs2">Blomgren, Peter; Chan, Tony F. (1998), "Color TV: total variation methods for restoration of vector-valued images", <i>IEEE Transactions on Image Processing</i>, <b>7</b> (3), Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309: <span class="nowrap">304–</span>309, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998ITIP....7..304B">1998ITIP....7..304B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F83.661180">10.1109/83.661180</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18276250">18276250</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Image+Processing&rft.atitle=Color+TV%3A+total+variation+methods+for+restoration+of+vector-valued+images&rft.volume=7&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E304-%3C%2Fspan%3E309&rft.date=1998&rft_id=info%3Apmid%2F18276250&rft_id=info%3Adoi%2F10.1109%2F83.661180&rft_id=info%3Abibcode%2F1998ITIP....7..304B&rft.aulast=Blomgren&rft.aufirst=Peter&rft.au=Chan%2C+Tony+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATotal+variation" class="Z3988"></span>.</li></ul> <ul><li><a href="/wiki/Tony_F._Chan" title="Tony F. Chan">Tony F. Chan</a> and Jackie (Jianhong) Shen (2005), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080117220948/http://jackieneoshen.googlepages.com/ImagingNewEra.html"><i>Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods</i></a>, <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-589-X" title="Special:BookSources/0-89871-589-X">0-89871-589-X</a> (with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).</li></ul> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐b766959bd‐krdpl Cached time: 20250214041449 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.558 seconds Real time usage: 0.872 seconds Preprocessor visited node count: 3552/1000000 Post‐expand include size: 83040/2097152 bytes Template argument size: 1751/2097152 bytes Highest expansion depth: 14/100 Expensive parser function count: 6/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 66175/5000000 bytes Lua time usage: 0.294/10.000 seconds Lua memory usage: 8149307/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 510.542 1 -total 20.40% 104.160 1 Template:Reflist 16.99% 86.749 2 Template:Cite_book 15.48% 79.029 1 Template:Short_description 13.41% 68.481 4 Template:Ambox 12.21% 62.347 1 Template:Primary_sources 11.81% 60.302 11 Template:Citation 11.31% 57.756 7 Template:Springer 10.30% 52.593 2 Template:Pagetype 7.21% 36.828 4 Template:Harv --> <!-- Saved in parser cache with key enwiki:pcache:683561:|#|:idhash:canonical and timestamp 20250214041449 and revision id 1268508491. Rendering was triggered because: page-view --> </div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Total_variation&oldid=1268508491">https://en.wikipedia.org/w/index.php?title=Total_variation&oldid=1268508491</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Mathematical_analysis" title="Category:Mathematical analysis">Mathematical analysis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_reliable_references_from_February_2012" title="Category:Articles lacking reliable references from February 2012">Articles lacking reliable references from February 2012</a></li><li><a href="/wiki/Category:All_articles_lacking_reliable_references" title="Category:All articles lacking reliable references">All articles lacking reliable references</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_September_2022" title="Category:Articles needing additional references from September 2022">Articles needing additional references from September 2022</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2012" title="Category:Articles needing additional references from May 2012">Articles needing additional references from May 2012</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_February_2012" title="Category:Articles lacking in-text citations from February 2012">Articles lacking in-text citations from February 2012</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:CS1_Italian-language_sources_(it)" title="Category:CS1 Italian-language sources (it)">CS1 Italian-language sources (it)</a></li><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Pages_that_use_a_deprecated_format_of_the_math_tags" title="Category:Pages that use a deprecated format of the math tags">Pages that use a deprecated format of the math tags</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 January 2025, at 03:35<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Total_variation&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" lang="en" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Total variation</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>10 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-qrf9m","wgBackendResponseTime":115,"wgPageParseReport":{"limitreport":{"cputime":"0.558","walltime":"0.872","ppvisitednodes":{"value":3552,"limit":1000000},"postexpandincludesize":{"value":83040,"limit":2097152},"templateargumentsize":{"value":1751,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":6,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":66175,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 510.542 1 -total"," 20.40% 104.160 1 Template:Reflist"," 16.99% 86.749 2 Template:Cite_book"," 15.48% 79.029 1 Template:Short_description"," 13.41% 68.481 4 Template:Ambox"," 12.21% 62.347 1 Template:Primary_sources"," 11.81% 60.302 11 Template:Citation"," 11.31% 57.756 7 Template:Springer"," 10.30% 52.593 2 Template:Pagetype"," 7.21% 36.828 4 Template:Harv"]},"scribunto":{"limitreport-timeusage":{"value":"0.294","limit":"10.000"},"limitreport-memusage":{"value":8149307,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAdamsClarkson1933\"] = 1,\n [\"CITEREFAmbrosioFuscoPallara2000\"] = 1,\n [\"CITEREFArzelà1905\"] = 1,\n [\"CITEREFBlomgrenChan1998\"] = 1,\n [\"CITEREFCasellesChambolleNovaga2007\"] = 1,\n [\"CITEREFCesari1936\"] = 1,\n [\"CITEREFGibbsFrancis_Edward_Su2002\"] = 1,\n [\"CITEREFGolubov\"] = 6,\n [\"CITEREFGolubovVitushkin\"] = 1,\n [\"CITEREFHahn1921\"] = 1,\n [\"CITEREFJordan1881\"] = 1,\n [\"CITEREFLeoni2017\"] = 1,\n [\"CITEREFRudin1966\"] = 1,\n [\"CITEREFRudinOsherFatemi1992\"] = 1,\n [\"CITEREFSaks1937\"] = 1,\n [\"CITEREFVitali1908\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Citation\"] = 11,\n [\"Citation needed section\"] = 1,\n [\"Cite book\"] = 2,\n [\"Cite web\"] = 1,\n [\"DEFAULTSORT:Total Variation\"] = 1,\n [\"Distinguish\"] = 1,\n [\"EquationNote\"] = 3,\n [\"EquationRef\"] = 6,\n [\"Harv\"] = 4,\n [\"Harvtxt\"] = 6,\n [\"Isbn\"] = 1,\n [\"Main\"] = 1,\n [\"MathWorld\"] = 1,\n [\"More footnotes\"] = 1,\n [\"PlanetMath\"] = 1,\n [\"Primary sources\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfn whitelist\"] = 1,\n [\"Short description\"] = 1,\n [\"Springer\"] = 7,\n [\"Unreferenced section\"] = 1,\n}\narticle_whitelist = table#1 {\n [\"CITEREFGolubovVitushkin2001\"] = 1,\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-b766959bd-krdpl","timestamp":"20250214041449","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Total variation","url":"https:\/\/en.wikipedia.org\/wiki\/Total_variation","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1936288","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1936288","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-05-27T23:10:04Z","dateModified":"2025-01-10T03:35:08Z","headline":"measure of the local oscillation behavior of a function"}</script> </body> </html>