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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">Cambiar a la tabla de contenidos</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">Conjunto rectificable</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="Ir a un artículo en otro idioma. 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href="https://en.wikipedia.org/wiki/Rectifiable_set" title="Rectifiable set (inglés)" lang="en" hreflang="en" data-title="Rectifiable set" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Suoristuva_joukko" title="Suoristuva joukko (finés)" lang="fi" hreflang="fi" data-title="Suoristuva joukko" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ensemble_rectifiable" title="Ensemble rectifiable (francés)" lang="fr" hreflang="fr" data-title="Ensemble rectifiable" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko 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</div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</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="es" dir="ltr"><p>Un <b>conjunto rectificable</b> es un conjunto que intuitivamente puede ser aproximado en el entorno de cada punto por un <a href="/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo">espacio euclídeo</a>. </p><p>Muchos <a href="/wiki/Objeto_matem%C3%A1tico" title="Objeto matemático">objetos matemáticos</a> definidos mediante <a href="/wiki/Aplicaci%C3%B3n_diferenciable" class="mw-redirect" title="Aplicación diferenciable">aplicaciones diferenciables</a> son rectificables (también llamados suaves). Mientras que los objetos <a href="/wiki/Fractal" title="Fractal">fractales</a> de aspecto irregular no suelen ser rectificables. </p> <div class="mw-heading mw-heading2"><h2 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunto_rectificable&action=edit&section=1" title="Editar sección: Definición"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un conjunto de <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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> es rectificable si en <a href="/wiki/Casi_en_todas_partes" title="Casi en todas partes">casi en todas partes</a> de su <a href="/wiki/Frontera_(topolog%C3%ADa)" title="Frontera (topología)">frontera topológica</a> admite un <a href="/wiki/Espacio_tangente" title="Espacio tangente">espacio tangente</a>. </p><p>Un <a href="/wiki/Subconjunto" title="Subconjunto">subconjunto</a> de <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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> es rectificable si dicho conjunto puede ser recubierto <a href="/wiki/Casi_en_todas_partes" title="Casi en todas partes">casi en todas partes</a> por una <a href="/wiki/Conjunto_numerable" title="Conjunto numerable">colección numerable</a> de piezas de Lipschitz <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_{1},E_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1},E_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43230653e777e537c4cb6c92d5cfce335f1db358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.33ex; height:2.509ex;" alt="{\displaystyle E_{1},E_{2},\dots }"></span> tales que: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><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 {H}}^{s}(E\setminus \cup _{i=1}^{n}E_{i})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>E</mi> <mo class="MJX-variant">∖</mo> <msubsup> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}^{s}(E\setminus \cup _{i=1}^{n}E_{i})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84f81be9688394699ed9cf2f5a887823c42c05b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.973ex; height:3.009ex;" alt="{\displaystyle {\mathcal {H}}^{s}(E\setminus \cup _{i=1}^{n}E_{i})=0}"></span> </p> </blockquote> <p>donde: </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 {\mathcal {H}}^{s}(\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}^{s}(\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38574b0d01df81895bea512bf89b8a6f9736e989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}^{s}(\cdot )}"></span> es la <a href="/wiki/Dimensi%C3%B3n_de_Hausdorff-Besicovitch#Medida_de_Hausdorff" title="Dimensión de Hausdorff-Besicovitch">medida de Hausdorff</a>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ejemplos">Ejemplos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conjunto_rectificable&action=edit&section=2" title="Editar sección: Ejemplos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Una <a href="/wiki/Circunferencia" title="Circunferencia">circunferencia</a> es un ejemplo de curva continua, cerrada y diferenciable que además es rectificable.</li> <li>La <a href="/wiki/Frontera_(topolog%C3%ADa)" title="Frontera (topología)">frontera</a> de un polígono cerrado es un ejemplo de curva continua, cerrada aunque no diferenciable en los vértices que aun así es rectificable, y por tanto dado un <a href="/wiki/Pol%C3%ADgono" title="Polígono">polígono</a> de <i>n</i> lados su perímetro tiene una longitud finita.</li> <li>La <a href="/wiki/Curva_de_Koch" class="mw-redirect" title="Curva de Koch">curva de Koch</a> es una curva no rectificable, su longitud no es finita, sin embargo encierra un área finita.</li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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