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id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="measurable_spaces">Measurable spaces</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#variations'>Variations</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToVonNeumannAlgebras'>Relation to von Neumann algebras</a></li> <li><a href='#relation_to_boolean_toposes'>Relation to Boolean toposes</a></li> </ul> <li><a href='#in_alternative_foundations'>In alternative foundations</a></li> <ul> <li><a href='#Predicative'>Predicative theory</a></li> <li><a href='#constructive_theory'>Constructive theory</a></li> <li><a href='#in_dream_mathematics'>In dream mathematics</a></li> <li><a href='#pointfree_measurable_spaces'>Point-free measurable spaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesRelationToVonNeumannAlgebras'>Relation to von Neumann algebra</a></li> <li><a href='#relation_to_boolean_toposes_2'>Relation to Boolean toposes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Measurable spaces are the traditional prelude to the general theory of <a class="existingWikiWord" href="/nlab/show/measure">measure</a> and <a class="existingWikiWord" href="/nlab/show/integration">integration</a>. Basically, a measure is a recipe for computing the size — e.g., length, area, volume — of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> of a given <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The structure of a ‘measurable space’ picks out those subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> for which the size is well-defined; these subsets are called ‘measurable’. The measure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is then an operation that assigns a number to each measurable subset saying how big it is.</p> <p>In short: you get a <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> by placing a measure on a measurable space.</p> <p>Ideally, all subsets would be measurable, but this contradicts the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> for the basic example of <a class="existingWikiWord" href="/nlab/show/Lebesgue+measure">Lebesgue measure</a> on the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>. Although it is possible to use nonstandard <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> of mathematics in which all subsets of the real line are Lebesgue measurable, any general theory that includes that example and is more general than those foundations requires some explicit notion of measurable space (or an alternative such as a <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a>).</p> <p>In any case, measurable spaces are of some interest in their own right, even without a measure on them.</p> <h2 id="definitions">Definitions</h2> <p>We give first the usual notion, assuming the validity of <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> and <a class="existingWikiWord" href="/nlab/show/power+sets">power sets</a>; see below for alternative versions, including the <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> and <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative</a> theories.</p> <p>Given a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/sigma-algebra">algebra</a></strong> is a collection of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that is closed under <a class="existingWikiWord" href="/nlab/show/complement">complementation</a>, <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> <a class="existingWikiWord" href="/nlab/show/unions">unions</a>, and countable <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>. A <strong>measurable space</strong>, by the usual modern definition, is a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. The elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> are called the <strong><a class="existingWikiWord" href="/nlab/show/measurable+sets">measurable sets</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (or more properly, the measurable subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\Sigma)</annotation></semantics></math>).</p> <p>Given measurable spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, a <strong><a class="existingWikiWord" href="/nlab/show/measurable+function">measurable function</a></strong> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^*(T)</annotation></semantics></math> is measurable in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is measurable in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. Measurable spaces and measurable functions form a <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Meas">Meas</a>, which is <a class="existingWikiWord" href="/nlab/show/topological+concrete+category">topological</a> over <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>In classical measure theory, it is usually assumed that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> (or a variation) equipped with the <a class="existingWikiWord" href="/nlab/show/Borel+sets">Borel sets</a> (see the examples below). Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is measurable if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(I)</annotation></semantics></math> is measurable whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">I \subseteq Y</annotation></semantics></math> is an interval.</p> <h2 id="variations">Variations</h2> <p>We will briefly examine variations of the notion of measurable space, from those most like the standard to those most unlike it. Most of these are discussed at articles dedicated to them.</p> <p>Historically, people have used more general notions that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebras, such as algebras, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math>-rings, and similar concepts whose names you can probably now guess; these are all discussed at <a class="existingWikiWord" href="/nlab/show/sigma-algebra">sigma-algebra</a>. These are all more general than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebras, being possibly not closed under some operations. When using some of these more general rings of measurable sets, it is necessary to allow <a class="existingWikiWord" href="/nlab/show/partial+functions">partial functions</a> whose domain is a <a class="existingWikiWord" href="/nlab/show/relatively+measurable+set">relatively measurable set</a> as measurable functions; for details, see <a class="existingWikiWord" href="/nlab/show/measurable+function">measurable function</a>.</p> <p>An <strong>enhanced measurable space</strong> has, in addition to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra of measurable sets, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/sigma-ideal">ideal</a> of <strong>measurable <a class="existingWikiWord" href="/nlab/show/null+sets">null sets</a></strong>. That is, besides the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-alebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, we have a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>⊆</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">N \subseteq \Sigma</annotation></semantics></math> that is closed under <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> unions and taking subsets (within <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>). (The elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> are the <em>measurable</em> null sets; a <strong><a class="existingWikiWord" href="/nlab/show/null+set">null set</a></strong> is <em>any</em> subset of a measurable null set.) One can equivalently specify a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/delta-filter">filter</a> of <strong>measurable <a class="existingWikiWord" href="/nlab/show/full+sets">full sets</a></strong>; the full sets are the <a class="existingWikiWord" href="/nlab/show/complements">complements</a> of the null sets. Either way, this allows us to use almost measurable <a class="existingWikiWord" href="/nlab/show/almost+functions">almost functions</a> up to <a class="existingWikiWord" href="/nlab/show/almost+equality">almost equality</a>, as described at <a class="existingWikiWord" href="/nlab/show/measurable+function">measurable function</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, because <a class="existingWikiWord" href="/nlab/show/complement">complementation</a> doesn't behave nicely, the concept of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra is not so useful. It's also essential to use <a class="existingWikiWord" href="/nlab/show/almost+functions">almost functions</a> to avoid a paucity of measurable functions. One solution, due to <a class="existingWikiWord" href="/nlab/show/Henry+Cheng">Henry Cheng</a>, may be found at <a class="existingWikiWord" href="/nlab/show/Cheng+measurable+space">Cheng measurable space</a>; briefly, we use <span class="newWikiWord">disjoint pairs<a href="/nlab/new/disjoint+pairs">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B)</annotation></semantics></math> of sets instead of individual measurable sets and use formal complements in the algebra, as well as a notion of <a class="existingWikiWord" href="/nlab/show/full+sets">full sets</a>. Assuming <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, a Cheng measurable space is actually equivalent to a measurable space equipped with null (or full) sets, as in the previous paragraph.</p> <p>In order to have the most important theorems of <a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a>, it is necessary and sufficient to restrict to <a class="existingWikiWord" href="/nlab/show/localizable+measures">localizable measures</a>. Since localizability refers only to the null (or full) sets, we can actually speak of a <strong><a class="existingWikiWord" href="/nlab/show/localizable+measurable+space">localizable measurable space</a></strong>: a measurable space equipped with null (or full) sets as above, with the property that the <a class="existingWikiWord" href="/nlab/show/boolean+algebra">boolean algebra</a> of measurable sets <a class="existingWikiWord" href="/nlab/show/quotient+algebra">modulo</a> the null sets is <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete</a>.</p> <p>Another approach to measure theory, more abstract, is to ignore the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and use <em>only</em> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, as an abstract <a class="existingWikiWord" href="/nlab/show/boolean+algebra">boolean algebra</a> equipped with <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> <a class="existingWikiWord" href="/nlab/show/suprema">suprema</a>; this is called a <strong><span class="newWikiWord">measurable algebra<a href="/nlab/new/measurable+algebra">?</a></span></strong> (or a <span class="newWikiWord">measure algebra<a href="/nlab/new/measure+algebra">?</a></span> when equipped with a <a class="existingWikiWord" href="/nlab/show/measure">measure</a>). A measurable algebra might also can be equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-ideal of null sets (or a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math>-filter of full sets), but really it is simpler to take the <a class="existingWikiWord" href="/nlab/show/quotient+algebra">quotient algebra</a>, which is itself a perfectly good measurable algebra. Even if a measurable algebra is a <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, it can still be pathological; but if it has enough <a class="existingWikiWord" href="/nlab/show/normal+measure">normal measure</a>s, then we have a <strong><a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></strong>; the <a class="existingWikiWord" href="/nlab/show/category">category</a> of measurable locales is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to that of <a class="existingWikiWord" href="/nlab/show/localizable+measurable+spaces">localizable measurable spaces</a> (from the previous paragraph).</p> <p>Yet another category equivalent to localizable measurable spaces and measurable locales is the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the category of <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative</a> <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebras">von Neumann algebras</a>; this is really a version of the <a class="existingWikiWord" href="/nlab/show/Gelfand%E2%80%93Neumark+theorem">Gelfand–Neumark theorem</a>. Then a <em><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative</a></em> (localizable) measurable space is (the formal dual of) <em>any</em> von Neumann algebra. In this way, measure theory may be seen as a branch of <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> theory (at least if one assumes that only localizable measurable spaces are well enough behaved to be worthy of study).</p> <h2 id="examples">Examples</h2> <ul> <li> <p>Of course, the <a class="existingWikiWord" href="/nlab/show/power+set">power set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is closed under all operations, so it is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra. Thus every set becomes a <strong><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete</a> measurable space</strong>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra generated by the open sets (or equivalently, by the closed sets) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <strong>Borel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra</strong>; its elements are called the <strong><a class="existingWikiWord" href="/nlab/show/Borel+sets">Borel sets</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. In particular, the Borel sets of <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>s are the Borel sets in the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> with its usual topology.</p> </li> <li> <p>If a measurable space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\Sigma)</annotation></semantics></math> is equipped with a measure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, making it into a <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a>, then the sets of measure zero form a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/sigma-ideal">ideal</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> (that is an <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> that is also a sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-ring). Let a <strong><a class="existingWikiWord" href="/nlab/show/null+subset">null set</a></strong> be <em>any</em> set (measurable or not) contained in a set of measure zero; then the null sets form a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-ideal in the <a class="existingWikiWord" href="/nlab/show/power+set">power set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Call a set <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>-measurable</strong> if it is the union of a measurable set and a null set; then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>-measurable sets form a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>μ</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_\mu</annotation></semantics></math> called the <strong>completion</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, and the measurable space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>Σ</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\Sigma_\mu)</annotation></semantics></math> is the <strong>completion</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\Sigma)</annotation></semantics></math>.</p> </li> <li> <p>In particular, the <strong>Lebesgue-measurable</strong> sets in the real line are the elements of the completion of the Borel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra under <a class="existingWikiWord" href="/nlab/show/Lebesgue+measure">Lebesgue measure</a>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="RelationToVonNeumannAlgebras">Relation to von Neumann algebras</h3> <p>One version of the <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality theorem</a> states that the category of <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/W%2A-algebra">algebras</a> is <a class="existingWikiWord" href="/nlab/show/dual+equivalence">dual</a> to the category of <em>compact strictly localizable</em> <a class="existingWikiWord" href="/nlab/show/enhanced+measurable+spaces">enhanced measurable spaces</a>. (As such, arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras may be interpreted as ‘noncommutative’ measurable spaces in a sense analogous to <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>.) See the <a href="#ReferencesRelationToVonNeumannAlgebras">references below</a>.</p> <p>To make this work correctly, we cannot simply define localizability as a <a class="existingWikiWord" href="/nlab/show/property">property</a> of measurable spaces; instead, a <strong><a class="existingWikiWord" href="/nlab/show/localizable+measurable+space">localizable measurable space</a></strong> is a measurable space (a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>) with a <a class="existingWikiWord" href="/nlab/show/%CF%83-ideal">σ-ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi></mrow><annotation encoding="application/x-tex">\mathcal{N}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}X</annotation></semantics></math> simultaneously (called the ideal of <strong>negligible sets</strong> or <strong><a class="existingWikiWord" href="/nlab/show/null+sets">null sets</a></strong>) such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">/</mo><mi>𝒩</mi></mrow><annotation encoding="application/x-tex">\Sigma/\mathcal{N}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>; and a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of localizable measurable spaces is a measurable function, with the property that the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of any null set is null, up to an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≅</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \cong g</annotation></semantics></math> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">m\in\Sigma</annotation></semantics></math> the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>m</mi><mo>⊕</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>m</mi></mrow><annotation encoding="application/x-tex">f^{-1}m\oplus g^{-1}m</annotation></semantics></math> is a null set. For good properties of point-set measurable maps, we need to add the property of <strong>strict localizability</strong> and <strong>compactness</strong>. See the article <a class="existingWikiWord" href="/nlab/show/categories+of+measure+theory">categories of measure theory</a> for full details.</p> <p>The requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">/</mo><mi>𝒩</mi></mrow><annotation encoding="application/x-tex">\Sigma/\mathcal{N}</annotation></semantics></math> be complete is the real localizability condition here; the trick of equipping a measurable space with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-ideal of null sets (or equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filter">filter</a> of <a class="existingWikiWord" href="/nlab/show/full+sets">full sets</a>) and taking measurable functions only up to equivalence is a common one in other situations.</p> <p>Localizable measurable spaces can also be studied via the lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">/</mo><mi>𝒩</mi></mrow><annotation encoding="application/x-tex">\Sigma/\mathcal{N}</annotation></semantics></math>, which is a <a class="existingWikiWord" href="/nlab/show/frame">frame</a>; the morphisms correspond to certain <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> between <a class="existingWikiWord" href="/nlab/show/locales">locales</a>, and thus we are studying locales with <a class="existingWikiWord" href="/nlab/show/extra+property">extra property</a>, called <strong><a class="existingWikiWord" href="/nlab/show/measurable+locales">measurable locales</a></strong>.</p> <h3 id="relation_to_boolean_toposes">Relation to Boolean toposes</h3> <p>In terms of <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>, measurable spaces are closely related to <a class="existingWikiWord" href="/nlab/show/Boolean+toposes">Boolean toposes</a> (e.g. <a href="#Jackson06">Jackson 06</a>, <a href="#Henry14">Henry 14</a>).</p> <h2 id="in_alternative_foundations">In alternative foundations</h2> <p>While Lebesgue measure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> can be done in very weak <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>, a general theory of measure and measurable spaces seems to require powerful <a class="existingWikiWord" href="/nlab/show/set+theory">set-theoretic</a> machinery. Indeed, not much seems to be possible in <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative</a> contexts, and the (nonpredicative) <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> theory is noticeably more complicated than the classical theory. On the other hand, the classical theory has its own complications, with nonmeasurable sets and functions that can be proved to exist but which seem to never arise in practice. Instead, there are classically false but apparently consistent foundations in which measure theory is extremely simple.</p> <h3 id="Predicative">Predicative theory</h3> <p>The main problem for measure theory in <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a> is getting your hands on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra. Once you've got that, you've got a measurable space (obviously) and go on to <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a>, where there are no new difficulties. However, what is (say) a Borel set in the real line? This is difficult, if not impossible, to explain predicatively. (In the case of <a class="existingWikiWord" href="/nlab/show/Lebesgue+measure">Lebesgue measure</a>, there <em>are</em> ways to describe the Lebesgue-measurable sets predicatively, but these do not seem to generalise to a broader theory.)</p> <p>Note that there is no real problem in describing what, say, an open set is. Not only can this be done for the real line in the usual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> way, but it is easy to take <em>any</em> collection of subsets of any set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, call that collection a <a class="existingWikiWord" href="/nlab/show/subbase">subbase</a>, and describe which sets are the open sets in the <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> generated by that subbase. The reason is that there are only two steps in moving from a subbase to a topology, and while the latter step is too impredicative to allow one to speak of the <em>set</em> of all open sets, it's OK if you only want to talk about <em>individual</em> open sets. (To be explicit: given a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to be used as subbase, a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is <em>open</em> if, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">x \in G</annotation></semantics></math>, then there exist a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_1, \ldots, A_n</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>A</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x \in A_i</annotation></semantics></math> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><msub><mi>A</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y \in A_i</annotation></semantics></math> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">y \in G</annotation></semantics></math>. Since we quantify only over points and natural numbers, not over sets or functions, this is a predicative definition, and it's easy to prove that the open sets satisfy the axioms of a topology.)</p> <p>This cannot be done with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebras, since we need uncountably many sets. To be sure, each individual step is predicative, and we can freely talk about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>δ</mi></msub></mrow><annotation encoding="application/x-tex">G_\delta</annotation></semantics></math> sets and the like, but to define a Borel set we need to quantify over all countable ordinals. While it is possible to hypothesise the existence of an uncountable ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1</annotation></semantics></math> and be predicative ‘over’ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1</annotation></semantics></math> (and after all, everything else in this section is only predicative over the first infinite ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\omega_0</annotation></semantics></math>, which we only have if we accept an <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a>), this cannot be constructed predicatively. (The immediate definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/Hartog%27s+number">Hartog's number</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\omega_0</annotation></semantics></math> uses <a class="existingWikiWord" href="/nlab/show/power+set">power set</a>s; while the construction of an uncountable ordinal by applying the <a class="existingWikiWord" href="/nlab/show/well-ordering+theorem">well-ordering theorem</a> to the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>N</mi></mstyle> <mstyle mathvariant="bold"><mi>N</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">\mathbf{N}^{\mathbf{N}}</annotation></semantics></math> doesn't seem to use reasoning that requires the existence of power sets as long as you don't also throw in <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, it does use reasoning that is not accepted by any predicative school that I know.)</p> <p>So as far as I (<a class="existingWikiWord" href="/nlab/show/Toby+Bartels">Toby Bartels</a>) can tell, there is no general predicative theory of measurable spaces, only an ad hoc theory of Lebsegue measurability. I would be delighted to learn otherwise!</p> <h3 id="constructive_theory">Constructive theory</h3> <p>From a constructive perspective, there are a couple of related problems with the classical theory. One is that the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra is highly suspicious, because it relies on an operation, <a class="existingWikiWord" href="/nlab/show/complement">complement</a>ation, that behaves very differently in the <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a> that <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> uses. The other is that, even you acept the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra anyway (after all, the Lebesgue-measurable sets on the real line do still form one), there may be very few measurable functions.</p> <p>Indeed, if we set aside the general theory of measurable spaces and simply do Lebesgue measure ad hoc in a constructive (even predicative) way, we find that instead of measurable <a class="existingWikiWord" href="/nlab/show/functions">functions</a> we really want measurable <a class="existingWikiWord" href="/nlab/show/partial+functions">partial functions</a> whose domain of definition is a <a class="existingWikiWord" href="/nlab/show/full+set">full set</a>. This suggests that if we want to define the concept of measurable function, then we have to know what the full sets are.</p> <p>There is a way out, due to <a class="existingWikiWord" href="/nlab/show/Henry+Cheng">Henry Cheng</a>, for both of these problems at once. Instead of dealing with individual sets, we will deal with pairs of <a class="existingWikiWord" href="/nlab/show/disjoint+sets">disjoint sets</a>. The intuition is that we use disjoint pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cup B</annotation></semantics></math> is full —with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo>¬</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\neg{A})</annotation></semantics></math> being the motivating example in the classical theory—, but we let the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra itself tell us which pairs those are. Once we fix a particular measure, we may find additional pairs whose union is full, somewhat like finding additional measurable sets when taking the completion in the classical theory (although taking the completion is a separate phenomenon here), but that's all right; the important thing is that each pair chosen really is full in any measure used (much as each set in a classical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra must actually be measurable by any measure used).</p> <p>See details at <a class="existingWikiWord" href="/nlab/show/Cheng+space">Cheng space</a>.</p> <p>There are also other approaches to constructive measure theory, such as <a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>‘s and <a class="existingWikiWord" href="/nlab/show/Erik+Palmgren">Erik Palmgren</a>’s notion of <span class="newWikiWord">metric Boolean algebra<a href="/nlab/new/metric+Boolean+algebra">?</a></span>.</p> <h3 id="in_dream_mathematics">In dream mathematics</h3> <p>While measure theory only gets more complicated in constructive mathematics, it becomes much easier in <a class="existingWikiWord" href="/nlab/show/dream+mathematics">dream mathematics</a>.</p> <p>… more coming …</p> <h3 id="pointfree_measurable_spaces">Point-free measurable spaces</h3> <p>Just like how in <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>, one takes the <a class="existingWikiWord" href="/nlab/show/opens">opens</a> to be fundamental and defines a <a class="existingWikiWord" href="/nlab/show/locale">locale</a> as a <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a>, in point-free measure theory, one takes the measurables to be fundamental and define the point-free analogue of a measurable space as a <a class="existingWikiWord" href="/nlab/show/sigma-complete+Boolean+algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-complete Boolean algebra</a> of measurables.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+measurable+space">noncommutative measurable space</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>For <a class="existingWikiWord" href="/nlab/show/Henry+Cheng">Henry Cheng</a>‘s theory of measure spaces, see</p> <ul> <li id="BishopBridges"> <p><a class="existingWikiWord" href="/nlab/show/Errett+Bishop">Errett Bishop</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Bridges">Douglas Bridges</a>: <em><a class="existingWikiWord" href="/nlab/show/Constructive+Analysis">Constructive Analysis</a></em>, Grundlehren der mathematischen Wissenschaften <strong>279</strong>, Springer (1985) [<a href="https://doi.org/10.1007/978-3-642-61667-9">doi:10.1007/978-3-642-61667-9</a>]</p> </li> <li id="BishopCheng"> <p><a class="existingWikiWord" href="/nlab/show/Errett+Bishop">Errett Bishop</a>, <a class="existingWikiWord" href="/nlab/show/Henry+Cheng">Henry Cheng</a>. <em>Constructive Measure Theory</em>, American Mathematical Society (1972). [ISBN:978-0821818169]</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>‘s and <a class="existingWikiWord" href="/nlab/show/Erik+Palmgren">Erik Palmgren</a>’s constructive measure theory, see</p> <ul> <li id="CoquandPalmgren"><a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>, <a class="existingWikiWord" href="/nlab/show/Erik+Palmgren">Erik Palmgren</a>, <em>Metric Boolean algebras and constructive measure theory</em>, Archive for Mathematical Logic 41, 687–704 (2002). [<a href="https://doi.org/10.1007/s001530100123">doi:10.1007/s001530100123</a>]</li> </ul> <h3 id="ReferencesRelationToVonNeumannAlgebras">Relation to von Neumann algebra</h3> <p>A discussion of the abstract properties of the category of <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebras">von Neumann algebras</a> is in</p> <ul> <li>Alain Guichardet, <em>Sur la catégorie des algèbres de von Neumann</em> (French) Bull. Sci. Math. (2) 90 1966 41–64. (<a href="http://ams.org/mathscinet-getitem?mr=201989">MSN</a>) (<a href="http://dmitripavlov.org/scans/guichardet.pdf">scan</a>)</li> </ul> <p>A modern treatment of this discussion can be found in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitri+Pavlov">Dmitri Pavlov</a>, <em>Gelfand-type duality for commutative von Neumann algebras</em>, Journal of Pure and Applied Algebra <strong>226</strong> 4 (2022) 106884 [<a href="https://arxiv.org/abs/2005.05284">arXiv:2005.05284</a>, <a href="https://doi.org/10.1016/j.jpaa.2021.106884">doi:10.1016/j.jpaa.2021.106884</a>]</li> </ul> <p>A useful series of expositions along these lines is in the article <a class="existingWikiWord" href="/nlab/show/categories+of+measure+theory">categories of measure theory</a>, as well as the MathOverflow posts</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitri+Pavlov">Dmitri Pavlov</a>, <em>On measurable spaces</em></p> <ul> <li> <p><a href="http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820">MO comment 1</a></p> </li> <li> <p><a href="http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542">MO comment 2</a></p> </li> </ul> </li> </ul> <p>See also</p> <ul> <li>Ryszard Paweł Kostecki, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^\ast</annotation></semantics></math>-algebras and noncommutative integration</em> (<a href="http://www.fuw.edu.pl/~kostecki/wstarint.pdf">pdf</a>)</li> </ul> <h3 id="relation_to_boolean_toposes_2">Relation to Boolean toposes</h3> <ul> <li id="Jackson06"> <p>Matthew Jackson, <em>A sheaf-theoretic approach to measure theory</em>, 2006 (<a href="http://www.andrew.cmu.edu/~awodey/students/jackson.pdf">pdf</a>)</p> </li> <li id="Henry14"> <p><a class="existingWikiWord" href="/nlab/show/Simon+Henry">Simon Henry</a>, <em>From toposes to non-commutative geometry through the study of internal Hilbert spaces</em>, 2014 (<a href="http://www.normalesup.org/~henry/Thesis.pdf">pdf</a>)</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/probability">probability</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on August 28, 2024 at 11:42:32. 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