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<title>Measurable function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Measurable function</span></span>

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</p><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, and in particular <a href="Mathematical_analysis#Measure_theory" title="Mathematical analysis">measure theory</a>, a <b>measurable function</b> is a function between the underlying sets of two <a href="Measurable_space" title="Measurable space">measurable spaces</a> that preserves the structure of the spaces: the <a href="Preimage" class="mw-redirect" title="Preimage">preimage</a> of any <a href="Measure_(mathematics)" title="Measure (mathematics)">measurable</a> set is measurable. This is in direct analogy to the definition that a <a href="Continuous_function" title="Continuous function">continuous</a> function between <a href="Topological_space" title="Topological space">topological spaces</a> <a href="Morphism" title="Morphism">preserves</a> the topological structure: the preimage of any <a href="Open_set" title="Open set">open set</a> is open. In <a href="Real_analysis" title="Real analysis">real analysis</a>, measurable functions are used in the definition of the <a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integral</a>. In <a href="Probability_theory" title="Probability theory">probability theory</a>, a measurable function on a <a href="Probability_space" title="Probability space">probability space</a> is known as a <a href="Random_variable" title="Random variable">random variable</a>.

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<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation>

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</math></span><img src="./63c4fcf15808c3dbf52cc4b3fd473e97e231f53f.svg" 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 )}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y,\mathrm {T} )}">

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<annotation encoding="application/x-tex">{\displaystyle (Y,\mathrm {T} )}</annotation>

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</math></span><img src="./38ad6b832e33b88ec69c140b994139d180c747e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.295ex; height:2.843ex;" alt="{\displaystyle (Y,\mathrm {T} )}" loading="lazy"></span> be measurable spaces, meaning that <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> are sets equipped with respective <a href="%CE%A3-algebra" title="Σ-algebra"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span>-algebras</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" 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 \Sigma }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {T} .}</annotation>

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</math></span><img src="./62cd2ca7157c8ae9fcf10598339b774c8294d5ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} .}" loading="lazy"></span> A function <span class="mwe-math-element mwe-math-element-inline"><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:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

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</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is said to be measurable if for every <span class="mwe-math-element mwe-math-element-inline"><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\in \mathrm {T} }">

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<annotation encoding="application/x-tex">{\displaystyle E\in \mathrm {T} }</annotation>

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</math></span><img src="./1561363cb166f3b821c651d37c0da466481af28a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.294ex; height:2.176ex;" alt="{\displaystyle E\in \mathrm {T} }" loading="lazy"></span> the pre-image of <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" 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}" loading="lazy"></span> under <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> is in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" 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 \Sigma }" loading="lazy"></span>; that is, for all <span class="mwe-math-element mwe-math-element-inline"><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\in \mathrm {T} }">

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<annotation encoding="application/x-tex">{\displaystyle E\in \mathrm {T} }</annotation>

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</math></span><img src="./1561363cb166f3b821c651d37c0da466481af28a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.294ex; height:2.176ex;" alt="{\displaystyle E\in \mathrm {T} }" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(E):=\{x\in X\mid f(x)\in E\}\in \Sigma .}</annotation>

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</p><p>That is, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (f)\subseteq \Sigma ,}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (f)\subseteq \Sigma ,}</annotation>

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</math></span><img src="./77e9c6d59bca8b4cd73eb9ed4f199b671f25b969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.841ex; height:2.843ex;" alt="{\displaystyle \sigma (f)\subseteq \Sigma ,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (f)}</annotation>

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</math></span><img src="./c2bda64d8dd43b33931f0f25adb128f42679e0ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle \sigma (f)}" loading="lazy"></span> is the <a href="%CE%A3-algebra#σ-algebra_generated_by_a_function" title="Σ-algebra">σ-algebra generated by f</a>. If <span class="mwe-math-element mwe-math-element-inline"><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:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

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</math></span><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"></span> is a measurable function, one writes

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<annotation encoding="application/x-tex">{\displaystyle f\colon (X,\Sigma )\rightarrow (Y,\mathrm {T} ).}</annotation>

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to emphasize the dependency on the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span>-algebras <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" 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 \Sigma }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {T} .}</annotation>

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</math></span><img src="./62cd2ca7157c8ae9fcf10598339b774c8294d5ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} .}" loading="lazy"></span>

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<div class="mw-heading mw-heading2"><h2 id="Term_usage_variations">Term usage variations</h2></div>

<p>The choice of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span>-algebras in the definition above is sometimes implicit and left up to the context. For example, for <span class="mwe-math-element mwe-math-element-inline"><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} ,}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation>

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</math></span><img src="./0522388d36b55de7babe4bbfc49475eaf590c2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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 {C} ,}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation>

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</math></span><img src="./c6ff6a3dc2982018ff20f1d2c927afc74a217be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" loading="lazy"></span> or other topological spaces, the <a href="Borel_algebra" class="mw-redirect" title="Borel algebra">Borel algebra</a> (generated by all the open sets) is a common choice. Some authors define <b>measurable functions</b> as exclusively real-valued ones with respect to the Borel algebra.<sup id="cite_ref-strichartz_1-0" class="reference"><a href="#cite_note-strichartz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>If the values of the function lie in an <a href="Infinite-dimensional_vector_space" class="mw-redirect" title="Infinite-dimensional vector space">infinite-dimensional vector space</a>, other non-equivalent definitions of measurability, such as <a href="Weak_measurability" class="mw-redirect" title="Weak measurability">weak measurability</a> and <a href="Bochner_measurability" class="mw-redirect" title="Bochner measurability">Bochner measurability</a>, exist.

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<div class="mw-heading mw-heading2"><h2 id="Notable_classes_of_measurable_functions">Notable classes of measurable functions</h2></div>

<ul><li>Random variables are by definition measurable functions defined on probability spaces.</li>

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<annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation>

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</math></span><img src="./63c4fcf15808c3dbf52cc4b3fd473e97e231f53f.svg" 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 )}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y,T)}">

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<annotation encoding="application/x-tex">{\displaystyle (Y,T)}</annotation>

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</math></span><img src="./80a999227d06f521df717822d93e55f38b60c117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.253ex; height:2.843ex;" alt="{\displaystyle (Y,T)}" loading="lazy"></span> are <a href="Borel_set#Standard_Borel_spaces_and_Kuratowski_theorems" title="Borel set">Borel spaces</a>, a measurable function <span class="mwe-math-element mwe-math-element-inline"><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:(X,\Sigma )\to (Y,T)}">

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<annotation encoding="application/x-tex">{\displaystyle f:(X,\Sigma )\to (Y,T)}</annotation>

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</math></span><img src="./ce6f9fb30416a16d5cd15505ecd6d4af3c3658f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.584ex; height:2.843ex;" alt="{\displaystyle f:(X,\Sigma )\to (Y,T)}" loading="lazy"></span> is also called a <b>Borel function</b>. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see <a href="Luzin's_theorem" class="mw-redirect" title="Luzin's theorem">Luzin's theorem</a>. If a Borel function happens to be a section of a map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\xrightarrow {~\pi ~} X,}">

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<annotation encoding="application/x-tex">{\displaystyle Y\xrightarrow {~\pi ~} X,}</annotation>

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</math></span><img src="./061b245aa5723b4d048cb5dcbce29404161d3135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.409ex; width:9.117ex; height:3.676ex;" alt="{\displaystyle Y\xrightarrow {~\pi ~} X,}" loading="lazy"></span> it is called a <b>Borel section</b>.</li>

<li>A <a href="Lebesgue_measurable" class="mw-redirect" title="Lebesgue measurable">Lebesgue measurable</a> function is a measurable function <span class="mwe-math-element mwe-math-element-inline"><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:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} }),}">

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<annotation encoding="application/x-tex">{\displaystyle f:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} }),}</annotation>

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</math></span><img src="./b8965b1fb86a1596c07924f561d769829f7ac960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.069ex; height:2.843ex;" alt="{\displaystyle f:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} }),}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><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 {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math></span><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"></span> is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span>-algebra of Lebesgue measurable sets, and <span class="mwe-math-element mwe-math-element-inline"><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 {B}}_{\mathbb {C} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{\mathbb {C} }}</annotation>

</semantics>

</math></span><img src="./304a777869e05d5abdc2c23b9fb26627f677b7dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.946ex; height:2.509ex;" alt="{\displaystyle {\mathcal {B}}_{\mathbb {C} }}" loading="lazy"></span> is the <a href="Borel_algebra" class="mw-redirect" title="Borel algebra">Borel algebra</a> on the <a href="Complex_number" title="Complex number">complex numbers</a> <span class="mwe-math-element mwe-math-element-inline"><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 {C} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" loading="lazy"></span> Lebesgue measurable functions are of interest in <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> because they can be integrated. In the case <span class="mwe-math-element mwe-math-element-inline"><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:X\to \mathbb {R} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {R} ,}</annotation>

</semantics>

</math></span><img src="./e72b93833f90a2d703ef1c01556e957bb187f351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.135ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} ,}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> is Lebesgue measurable if and only if <span class="mwe-math-element mwe-math-element-inline"><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>\alpha \}=\{x\in X:f(x)>\alpha \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f>\alpha \}=\{x\in X:f(x)>\alpha \}}</annotation>

</semantics>

</math></span><img src="./25552d6674eb860bff75240d30fe7f1baf3596fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.704ex; height:2.843ex;" alt="{\displaystyle \{f>\alpha \}=\{x\in X:f(x)>\alpha \}}" loading="lazy"></span> is measurable for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {R} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {R} .}</annotation>

</semantics>

</math></span><img src="./1f6e048324920f574b8d43e3b5e45063aafc7fd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.653ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {R} .}" loading="lazy"></span> This is also equivalent to any of <span class="mwe-math-element mwe-math-element-inline"><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\geq \alpha \},\{f<\alpha \},\{f\leq \alpha \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f\geq \alpha \},\{f<\alpha \},\{f\leq \alpha \}}</annotation>

</semantics>

</math></span><img src="./7b07f70a278471ca9dd0198bd082f57d7110bca4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.637ex; height:2.843ex;" alt="{\displaystyle \{f\geq \alpha \},\{f<\alpha \},\{f\leq \alpha \}}" loading="lazy"></span> being measurable for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,}</annotation>

</semantics>

</math></span><img src="./4b2cc8f6d373595f06dcd33f127dadf2b9d5727f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" alt="{\displaystyle \alpha ,}" loading="lazy"></span> or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.<sup id="cite_ref-carothers_2-0" class="reference"><a href="#cite_note-carothers-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A function <span class="mwe-math-element mwe-math-element-inline"><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:X\to \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./a57fe120f564b03d50ef328f6a5e09d0e860781e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {C} }" loading="lazy"></span> is measurable if and only if the real and imaginary parts are measurable.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Properties_of_measurable_functions">Properties of measurable functions</h2></div>

<ul><li>The sum and product of two complex-valued measurable functions are measurable.<sup id="cite_ref-folland_3-0" class="reference"><a href="#cite_note-folland-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> So is the quotient, so long as there is no division by zero.<sup id="cite_ref-strichartz_1-1" class="reference"><a href="#cite_note-strichartz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}</annotation>

</semantics>

</math></span><img src="./7c79f5d54e682f6bfc3361a779908a8fd80edf2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.734ex; height:2.843ex;" alt="{\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:(Y,\Sigma _{2})\to (Z,\Sigma _{3})}">

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:(Y,\Sigma _{2})\to (Z,\Sigma _{3})}</annotation>

</semantics>

</math></span><img src="./e904ea3425594c3ca1a5f9ee58dcec103e4766ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.272ex; height:2.843ex;" alt="{\displaystyle g:(Y,\Sigma _{2})\to (Z,\Sigma _{3})}" loading="lazy"></span> are measurable functions, then so is their composition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:(X,\Sigma _{1})\to (Z,\Sigma _{3}).}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f:(X,\Sigma _{1})\to (Z,\Sigma _{3}).}</annotation>

</semantics>

</math></span><img src="./38d8a1976b00902f21fa26111c1cdb24422a91b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.599ex; height:2.843ex;" alt="{\displaystyle g\circ f:(X,\Sigma _{1})\to (Z,\Sigma _{3}).}" loading="lazy"></span><sup id="cite_ref-strichartz_1-2" class="reference"><a href="#cite_note-strichartz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}</annotation>

</semantics>

</math></span><img src="./7c79f5d54e682f6bfc3361a779908a8fd80edf2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.734ex; height:2.843ex;" alt="{\displaystyle f:(X,\Sigma _{1})\to (Y,\Sigma _{2})}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:(Y,\Sigma _{3})\to (Z,\Sigma _{4})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:(Y,\Sigma _{3})\to (Z,\Sigma _{4})}</annotation>

</semantics>

</math></span><img src="./898f3d5e433f25d3a52f1a5a5d965da95a9ffb89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.272ex; height:2.843ex;" alt="{\displaystyle g:(Y,\Sigma _{3})\to (Z,\Sigma _{4})}" loading="lazy"></span> are measurable functions, their composition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:X\to Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f:X\to Z}</annotation>

</semantics>

</math></span><img src="./5e5cfd853da7199884ac0460cecd0b281f9846d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.801ex; height:2.509ex;" alt="{\displaystyle g\circ f:X\to Z}" loading="lazy"></span> need not be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Sigma _{1},\Sigma _{4})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Sigma _{1},\Sigma _{4})}</annotation>

</semantics>

</math></span><img src="./292d96be03dcfba7b221cf8ba2e2b9333883d5b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.308ex; height:2.843ex;" alt="{\displaystyle (\Sigma _{1},\Sigma _{4})}" loading="lazy"></span>-measurable unless <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{3}\subseteq \Sigma _{2}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma _{3}\subseteq \Sigma _{2}.}</annotation>

</semantics>

</math></span><img src="./c4aa9277e07a4604061b76f4f9dbcd8dd0d27c18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.21ex; height:2.509ex;" alt="{\displaystyle \Sigma _{3}\subseteq \Sigma _{2}.}" loading="lazy"></span> Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.</li>

<li>The (pointwise) <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a>, <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a>, <a href="Limit_superior" class="mw-redirect" title="Limit superior">limit superior</a>, and <a href="Limit_inferior" class="mw-redirect" title="Limit inferior">limit inferior</a> of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.<sup id="cite_ref-strichartz_1-3" class="reference"><a href="#cite_note-strichartz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-royden_4-0" class="reference"><a href="#cite_note-royden-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li>

<li>The <a href="Pointwise" title="Pointwise">pointwise</a> limit of a sequence of measurable functions <span class="mwe-math-element mwe-math-element-inline"><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_{n}:X\to Y}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f_{n}:X\to Y}</annotation>

</semantics>

</math></span><img src="./900c3c662c67bdbfe5055e20f54d411cd039578b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.662ex; height:2.509ex;" alt="{\displaystyle f_{n}:X\to Y}" loading="lazy"></span> is measurable, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> is a metric space (endowed with the Borel algebra). This is not true in general if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> is non-metrizable. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.<sup id="cite_ref-dudley_5-0" class="reference"><a href="#cite_note-dudley-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-aliprantis_6-0" class="reference"><a href="#cite_note-aliprantis-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Non-measurable_functions">Non-measurable functions</h2></div>

<p>Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> in an essential way, in the sense that <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> without the axiom of choice does not prove the existence of such functions.

</p><p>In any measure space <i><span class="mwe-math-element mwe-math-element-inline"><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 )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation>

</semantics>

</math></span><img src="./63c4fcf15808c3dbf52cc4b3fd473e97e231f53f.svg" 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 )}" loading="lazy"></span></i> with a <a href="Non-measurable_set" title="Non-measurable set">non-measurable set</a> <span class="mwe-math-element mwe-math-element-inline"><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\subset X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subset X,}</annotation>

</semantics>

</math></span><img src="./2dfb676b926f87c10788d964dcf7bf92032e5352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subset X,}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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\notin \Sigma ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\notin \Sigma ,}</annotation>

</semantics>

</math></span><img src="./bb0e09543e9d2d05d808517ab871310fc1c17425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.909ex; height:2.676ex;" alt="{\displaystyle A\notin \Sigma ,}" loading="lazy"></span> one can construct a non-measurable <a href="Indicator_function" title="Indicator function">indicator function</a>:

<msub>

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<msub>

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<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> otherwise</mtext>

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</mtd>

</mtr>

</mtable>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{A}:(X,\Sigma )\to \mathbb {R} ,\quad \mathbf {1} _{A}(x)={\begin{cases}1&{\text{ if }}x\in A\\0&{\text{ otherwise}},\end{cases}}}</annotation>

</semantics>

</math></span></span>

where <span class="mwe-math-element mwe-math-element-inline"><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} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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 \mathbb {R} }" loading="lazy"></span> is equipped with the usual <a href="Borel_algebra" class="mw-redirect" title="Borel algebra">Borel algebra</a>. This is a non-measurable function since the preimage of the measurable set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1\}}</annotation>

</semantics>

</math></span><img src="./5acdcac635f883f8b4f0a01aa03b16b22f23b124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}" loading="lazy"></span> is the non-measurable <span class="mwe-math-element mwe-math-element-inline"><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.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A.}</annotation>

</semantics>

</math></span><img src="./8a71bf21ad35b8fe05555041d54d1e17eeb0f490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.39ex; height:2.176ex;" alt="{\displaystyle A.}" loading="lazy"></span>  

</p><p>As another example, any non-constant function <span class="mwe-math-element mwe-math-element-inline"><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:X\to \mathbb {R} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {R} }</annotation>

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</math></span><img src="./669fa4832da4b0b229d77eadb270e95188f2eb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} }" loading="lazy"></span> is non-measurable with respect to the trivial <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span>-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma =\{\varnothing ,X\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma =\{\varnothing ,X\},}</annotation>

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</math></span><img src="./f6cd353683485c4994d1cc97ddf77c56fb495b13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.57ex; height:2.843ex;" alt="{\displaystyle \Sigma =\{\varnothing ,X\},}" loading="lazy"></span> since the preimage of any point in the range is some proper, nonempty subset of <span class="mwe-math-element mwe-math-element-inline"><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,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

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</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> which is not an element of the trivial <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma .}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma .}</annotation>

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</math></span><img src="./203943be81c35e85bc0f08a1f8822bf725dc43a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \Sigma .}" loading="lazy"></span>

</p>

<ul><li><a href="Bochner_measurable_function" title="Bochner measurable function">Bochner measurable function</a></li>

<li><a href="Bochner_space" title="Bochner space">Bochner space</a> – Type of topological space</li>

<li><a href="Lp_space" title="Lp space">Lp space</a> – Function spaces generalizing finite-dimensional p norm spaces - Vector spaces of measurable functions: the <a href="Lp_space" title="Lp space"><span class="mwe-math-element mwe-math-element-inline"><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^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation>

</semantics>

</math></span><img src="./cf2317aaca1ecee4b8ccf667bc1001059eae5850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}" loading="lazy"></span> spaces</a></li>

<li><a href="Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">Measure-preserving dynamical system</a> – Subject of study in ergodic theory</li>

<li><a href="Vector_measure" title="Vector measure">Vector measure</a></li>

<li><a href="Weakly_measurable_function" title="Weakly measurable function">Weakly measurable function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</div>

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</style><cite id="CITEREFStrichartz2000" class="citation book cs1">Strichartz, Robert (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/wayofanalysis0000stri"><i>The Way of Analysis</i></a></span>. Jones and Bartlett. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7637-1497-6</bdi>.</cite></span>

</li>

<li id="cite_note-carothers-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-carothers_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCarothers2000" class="citation book cs1">Carothers, N. L. (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/realanalysis0000caro"><i>Real Analysis</i></a></span>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-49756-6</bdi>.</cite></span>

</li>

<li id="cite_note-folland-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-folland_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFolland1999" class="citation book cs1">Folland, Gerald B. (1999). <i>Real Analysis: Modern Techniques and their Applications</i>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-31716-0</bdi>.</cite></span>

</li>

<li id="cite_note-royden-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-royden_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoyden1988" class="citation book cs1">Royden, H. L. (1988). <i>Real Analysis</i>. Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-02-404151-3</bdi>.</cite></span>

</li>

<li id="cite_note-dudley-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-dudley_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDudley2002" class="citation book cs1">Dudley, R. M. (2002). <i>Real Analysis and Probability</i> (2 ed.). Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-00754-2</bdi>.</cite></span>

</li>

<li id="cite_note-aliprantis-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-aliprantis_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAliprantisBorder2006" class="citation book cs1">Aliprantis, Charalambos D.; Border, Kim C. (2006). <i>Infinite Dimensional Analysis, A Hitchhiker's Guide</i> (3 ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-29587-7</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Measurable_function">Measurable function</a> at <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></li>

<li><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Borel_function">Borel function</a> at <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></li></ul>

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</style><div id="Measure_theory138" style="font-size:114%;margin:0 4em"><a href="Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_continuity" title="Absolute continuity">Absolute continuity</a> <a href="Absolute_continuity_(measure_theory)" class="mw-redirect" title="Absolute continuity (measure theory)">of measures</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a>

<ul><li><a href="Probability_space" title="Probability space">Probability space</a></li></ul></li>

<li><a href="Measurable_space" title="Measurable space">Measurable space</a>/</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li>

<li><a href="Baire_set" title="Baire set">Baire set</a></li>

<li><a href="Borel_set" title="Borel set">Borel set</a>

<ul><li><a href="Borel_equivalence_relation" title="Borel equivalence relation">equivalence relation</a></li></ul></li>

<li><a href="Standard_Borel_space" title="Standard Borel space">Borel space</a></li>

<li><a href="Carath%C3%A9odory's_criterion" title="Carathéodory's criterion">Carathéodory's criterion</a></li>

<li><a href="Cylindrical_%CF%83-algebra" title="Cylindrical σ-algebra">Cylindrical σ-algebra</a>

<ul><li><a href="Cylinder_set" title="Cylinder set">Cylinder set</a></li></ul></li>

<li><a href="Dynkin_system" title="Dynkin system">𝜆-system</a></li>

<li><a href="Essential_range" title="Essential range">Essential range</a>

<ul><li><a href="Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">infimum/supremum</a></li></ul></li>

<li><a href="Locally_measurable_set" class="mw-redirect" title="Locally measurable set">Locally measurable</a></li>

<li><a href="Pi-system" title="Pi-system"><span class="texhtml mvar" style="font-style:italic;">π</span>-system</a></li>

<li><a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a></li>

<li><a href="Non-measurable_set" title="Non-measurable set">Non-measurable set</a>

<ul><li><a href="Vitali_set" title="Vitali set">Vitali set</a></li></ul></li>

<li><a href="Support_(measure_theory)" title="Support (measure theory)">Support</a></li>

<li><a href="Transverse_measure" title="Transverse measure">Transverse measure</a></li>

<li><a href="Universally_measurable_set" title="Universally measurable set">Universally measurable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Measure_(mathematics)" title="Measure (mathematics)">measures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Atomic_measure" class="mw-redirect" title="Atomic measure">Atomic</a></li>

<li><a href="Baire_measure" title="Baire measure">Baire</a></li>

<li><a href="Banach_measure" title="Banach measure">Banach</a></li>

<li><a href="Besov_measure" title="Besov measure">Besov</a></li>

<li><a href="Borel_measure" title="Borel measure">Borel</a></li>

<li><a href="Brown_measure" title="Brown measure">Brown</a></li>

<li><a href="Complex_measure" title="Complex measure">Complex</a></li>

<li><a href="Complete_measure" title="Complete measure">Complete</a></li>

<li><a href="Content_(measure_theory)" title="Content (measure theory)">Content</a></li>

<li>(<a href="Logarithmically_concave_measure" title="Logarithmically concave measure">Logarithmically</a>) <a href="Convex_measure" title="Convex measure">Convex</a></li>

<li><a href="Decomposable_measure" title="Decomposable measure">Decomposable</a></li>

<li><a href="Discrete_measure" title="Discrete measure">Discrete</a></li>

<li><a href="Equivalence_(measure_theory)" title="Equivalence (measure theory)">Equivalent</a></li>

<li><a href="Finite_measure" title="Finite measure">Finite</a></li>

<li><a href="Inner_measure" title="Inner measure">Inner</a></li>

<li>(<a href="Quasi-invariant_measure" title="Quasi-invariant measure">Quasi-</a>) <a href="Invariant_measure" title="Invariant measure">Invariant</a></li>

<li><a href="Locally_finite_measure" title="Locally finite measure">Locally finite</a></li>

<li><a href="Maximising_measure" title="Maximising measure">Maximising</a></li>

<li><a href="Metric_outer_measure" title="Metric outer measure">Metric outer</a></li>

<li><a href="Outer_measure" title="Outer measure">Outer</a></li>

<li><a href="Perfect_measure" title="Perfect measure">Perfect</a></li>

<li><a href="Pre-measure" title="Pre-measure">Pre-measure</a></li>

<li>(<a href="Sub-probability_measure" title="Sub-probability measure">Sub-</a>) <a href="Probability_measure" title="Probability measure">Probability</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li>

<li><a href="Radon_measure" title="Radon measure">Radon</a></li>

<li><a href="Random_measure" title="Random measure">Random</a></li>

<li><a href="Regular_measure" title="Regular measure">Regular</a>

<ul><li><a href="Borel_regular_measure" title="Borel regular measure">Borel regular</a></li>

<li><a href="Inner_regular_measure" class="mw-redirect" title="Inner regular measure">Inner regular</a></li>

<li><a href="Outer_regular_measure" class="mw-redirect" title="Outer regular measure">Outer regular</a></li></ul></li>

<li><a href="Saturated_measure" title="Saturated measure">Saturated</a></li>

<li><a href="Set_function" title="Set function">Set function</a></li>

<li><a href="%CE%A3-finite_measure" title="Σ-finite measure">σ-finite</a></li>

<li><a href="S-finite_measure" title="S-finite measure">s-finite</a></li>

<li><a href="Signed_measure" title="Signed measure">Signed</a></li>

<li><a href="Singular_measure" title="Singular measure">Singular</a></li>

<li><a href="Spectral_measure" class="mw-redirect" title="Spectral measure">Spectral</a></li>

<li><a href="Strictly_positive_measure" title="Strictly positive measure">Strictly positive</a></li>

<li><a href="Tightness_of_measures" title="Tightness of measures">Tight</a></li>

<li><a href="Vector_measure" title="Vector measure">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Particular measures</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Counting_measure" title="Counting measure">Counting</a></li>

<li><a href="Dirac_measure" title="Dirac measure">Dirac</a></li>

<li><a href="Euler_measure" title="Euler measure">Euler</a></li>

<li><a href="Gaussian_measure" title="Gaussian measure">Gaussian</a></li>

<li><a href="Harmonic_measure" title="Harmonic measure">Harmonic</a></li>

<li><a href="Hausdorff_measure" title="Hausdorff measure">Hausdorff</a></li>

<li><a href="Intensity_measure" title="Intensity measure">Intensity</a></li>

<li><a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a>

<ul><li><a href="Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Infinite-dimensional</a></li></ul></li>

<li><a href="Positive_real_numbers#Logarithmic_measure" title="Positive real numbers">Logarithmic</a></li>

<li><a href="Product_measure" title="Product measure">Product</a>

<ul><li><a href="Projection_(measure_theory)" title="Projection (measure theory)">Projections</a></li></ul></li>

<li><a href="Pushforward_measure" title="Pushforward measure">Pushforward</a></li>

<li><a href="Spherical_measure" title="Spherical measure">Spherical measure</a></li>

<li><a href="Tangent_measure" title="Tangent measure">Tangent</a></li>

<li><a href="Trivial_measure" title="Trivial measure">Trivial</a></li>

<li><a href="Young_measure" title="Young measure">Young</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bochner_measurable_function" title="Bochner measurable function">Bochner</a></li>

<li><a href="Strongly_measurable_function" title="Strongly measurable function">Strongly</a></li>

<li><a href="Weakly_measurable_function" title="Weakly measurable function">Weakly</a></li></ul></li>

<li>Convergence: <a href="Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">almost everywhere</a></li>

<li><a href="Convergence_of_measures" title="Convergence of measures">of measures</a></li>

<li><a href="Convergence_in_measure" title="Convergence in measure">in measure</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">of random variables</a>

<ul><li><a href="Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">in distribution</a></li>

<li><a href="Convergence_in_probability" class="mw-redirect" title="Convergence in probability">in probability</a></li></ul></li>

<li><a href="Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li>

<li>Random: <a href="Random_compact_set" title="Random compact set">compact set</a></li>

<li><a href="Random_element" title="Random element">element</a></li>

<li><a href="Random_measure" title="Random measure">measure</a></li>

<li><a href="Stochastic_process" title="Stochastic process">process</a></li>

<li><a href="Random_variable" title="Random variable">variable</a></li>

<li><a href="Multivariate_random_variable" title="Multivariate random variable">vector</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carath%C3%A9odory's_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a></li>

<li>Convergence theorems

<ul><li><a href="Dominated_convergence_theorem" title="Dominated convergence theorem">Dominated</a></li>

<li><a href="Monotone_convergence_theorem" title="Monotone convergence theorem">Monotone</a></li>

<li><a href="Vitali_convergence_theorem" title="Vitali convergence theorem">Vitali</a></li></ul></li>

<li>Decomposition theorems

<li><a href="Jordan_decomposition_theorem" class="mw-redirect" title="Jordan decomposition theorem">Jordan</a></li>

<li><a href="Maharam's_theorem" title="Maharam's theorem">Maharam's</a></li></ul></li>

<li><a href="Egorov's_theorem" title="Egorov's theorem">Egorov's</a></li>

<li><a href="Fatou's_lemma" title="Fatou's lemma">Fatou's lemma</a></li>

<li><a href="Fubini's_theorem" title="Fubini's theorem">Fubini's</a>

<ul><li><a href="Fubini%E2%80%93Tonelli_theorem" class="mw-redirect" title="Fubini–Tonelli theorem">Fubini–Tonelli</a></li></ul></li>

<li><a href="H%C3%B6lder's_inequality" title="Hölder's inequality">Hölder's inequality</a></li>

<li><a href="Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a></li>

<li><a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym</a></li>

<li><a href="Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem" title="Riesz–Markov–Kakutani representation theorem">Riesz–Markov–Kakutani representation theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Disintegration_theorem" title="Disintegration theorem">Disintegration theorem</a>

<ul><li><a href="Lifting_theory" title="Lifting theory">Lifting theory</a></li></ul></li>

<li><a href="Lebesgue's_density_theorem" title="Lebesgue's density theorem">Lebesgue's density theorem</a></li>

<li><a href="Lebesgue_differentiation_theorem" title="Lebesgue differentiation theorem">Lebesgue differentiation theorem</a></li>

<li><a href="Sard's_theorem" title="Sard's theorem">Sard's theorem</a></li>

<li><a href="Vitali%E2%80%93Hahn%E2%80%93Saks_theorem" title="Vitali–Hahn–Saks theorem">Vitali–Hahn–Saks theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size: 85%;">For <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li>

<li><a href="Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a>

<ul><li><a href="Milman's_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li>

<li><a href="Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li>

<li><a href="Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li>

<li><a href="Vitale's_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_analysis" title="Convex analysis">Convex analysis</a></li>

<li><a href="Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Spectral_theory" title="Spectral theory">Spectral theory</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Lp_spaces64" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lp_spaces64" style="font-size:114%;margin:0 4em"><a href="Lp_space" title="Lp space">Lp spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_space" title="Banach space">Banach</a> & <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a></li>

<li><a href="Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a>

<ul><li><a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a></li></ul></li>

<li><a href="Measure_space" title="Measure space">Measure space</a></li>

<li><a href="Measurable_space" title="Measurable space">Measurable space</a>/</li>

<li><a href="Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li>

<li><a href="Sequence_space" title="Sequence space">Sequence spaces</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="L1_space" class="mw-redirect" title="L1 space"><i>L</i><sup>1</sup> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="L2_space" class="mw-redirect" title="L2 space"><i>L</i><sup>2</sup> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bessel's_inequality" title="Bessel's inequality">Bessel's</a></li>

<li><a href="Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li>

<li><a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert space</a></li>

<li><a href="Parseval's_identity" title="Parseval's identity">Parseval's identity</a></li>

<li><a href="Polarization_identity" title="Polarization identity">Polarization identity</a></li>

<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li>

<li><a href="Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="L-infinity" title="L-infinity"><span class="mwe-math-element mwe-math-element-inline"><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^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation>

</semantics>

</math></span><img src="./b9ab400cc4dfd865180cd84c72dc894ca457671f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}" loading="lazy"></span> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bounded_function" title="Bounded function">Bounded function</a></li>

<li><a href="Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li>

<li><a href="Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a>

<ul><li><a href="Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li>

<li><a href="Uniform_norm" title="Uniform norm">Uniform norm</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li>

<li><a href="Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li>

<li><a href="Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li>

<li><a href="Function_space" title="Function space">Function space</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li>

<li><a href="Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li>

<li><a href="Chebyshev's_inequality" title="Chebyshev's inequality">Chebyshev's</a></li>

<li><a href="Clarkson's_inequalities" title="Clarkson's inequalities">Clarkson's</a></li>

<li><a href="Hanner's_inequalities" title="Hanner's inequalities">Hanner's</a></li>

<li><a href="Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li>

<li><a href="H%C3%B6lder's_inequality" title="Hölder's inequality">Hölder's</a></li>

<li><a href="Markov's_inequality" title="Markov's inequality">Markov's</a></li>

<li><a href="Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li>

<li><a href="Young's_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li>

<li><a href="Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li>

<li><a href="Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li>

<li><a href="Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li>

<li><a href="Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size: 85%;">For <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li>

<li><a href="Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a>

<ul><li><a href="Milman's_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li>

<li><a href="Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li>

<li><a href="Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li>

<li><a href="Vitale's_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bochner_space" title="Bochner space">Bochner space</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Lorentz_space" title="Lorentz space">Lorentz space</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Quasinorm" title="Quasinorm">Quasinorm</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev space</a></li>

<li><a href="*-algebra" title="*-algebra">*-algebra</a>

<ul><li><a href="C*-algebra" title="C*-algebra">C*-algebra</a></li>

<li><a href="Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann</a></li></ul></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Function330" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

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