SherbyNet - /page/wikibook/entry.mu

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<title>Essential range</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Essential range</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, particularly <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, the <b>essential range</b>, or the set of <b>essential values</b>, of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>. One way of thinking of the essential range of a function is the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> on which the range of the function is 'concentrated'.

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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,{\cal {A}},\mu )}</annotation>

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</math></span><img src="./93c2231ecc1de29bbf69188c03ffa89dd217f085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.843ex;" alt="{\displaystyle (X,{\cal {A}},\mu )}" loading="lazy"></span> be a <a href="Measure_space" title="Measure space">measure space</a>, and let <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,{\cal {T}})}">

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

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</math></span><img src="./26e23dc2d90da82d440a741a9793053c35c04d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.552ex; height:2.843ex;" alt="{\displaystyle (Y,{\cal {T}})}" loading="lazy"></span> be a <a href="Topological_space" title="Topological space">topological space</a>. For any <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 ({\cal {A}},\sigma ({\cal {T}}))}">

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

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</math></span><img src="./4733ea6ad875aa2f051e05dd3f5fa747b6704352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.821ex; height:2.843ex;" alt="{\displaystyle ({\cal {A}},\sigma ({\cal {T}}))}" loading="lazy"></span>-<a href="Measurable_function" title="Measurable function">measurable function</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 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>, we say the <b>essential range</b> 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}">

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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> to mean the set

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.}</annotation>

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</math></span><img src="./f734db9dbdb010080e66c5e3fcb7622790745910.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.841ex; height:3.343ex;" alt="{\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.}" loading="lazy"></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Example 0.A.5">: Example 0.A.5 </span></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></dd></dl>

<p>Equivalently, <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 \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}</annotation>

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</math></span><img src="./f34a66679fcb34453ca96547360ae878bfa1aee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.866ex; height:2.843ex;" alt="{\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}" 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 f_{*}\mu }">

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<annotation encoding="application/x-tex">{\displaystyle f_{*}\mu }</annotation>

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</math></span><img src="./c84eeadae4cbdb2162050c53d5a759f92fc2ae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.595ex; height:2.676ex;" alt="{\displaystyle f_{*}\mu }" loading="lazy"></span> is the <a href="Pushforward_measure" title="Pushforward measure">pushforward measure</a> onto <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 ({\cal {T}})}">

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

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</math></span><img src="./c1d3344d197de3c4d55a6c1c58b77d350244d981.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.075ex; height:2.843ex;" alt="{\displaystyle \sigma ({\cal {T}})}" loading="lazy"></span> 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 \mu }">

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

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</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" 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> 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 \operatorname {supp} (f_{*}\mu )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {supp} (f_{*}\mu )}</annotation>

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</math></span><img src="./d044fa26596ebe5f5ac31528c1fb752f75f8088f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.198ex; height:2.843ex;" alt="{\displaystyle \operatorname {supp} (f_{*}\mu )}" loading="lazy"></span> denotes the <a href="Support_(measure_theory)" title="Support (measure theory)">support</a> 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_{*}\mu .}">

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

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</math></span><img src="./bcca174df3f380c66d351467178f67f8cde38d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.242ex; height:2.676ex;" alt="{\displaystyle f_{*}\mu .}" loading="lazy"></span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Essential_values">Essential values</h3></div>

<p>The phrase "<b>essential value</b> 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}">

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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 sometimes used to mean an element of the essential range 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.}">

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

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</math></span><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"></span><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Exercise 4.1.6">: Exercise 4.1.6 </span></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Example 7.1.11">: Example 7.1.11 </span></sup>

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<div class="mw-heading mw-heading2"><h2 id="Special_cases_of_common_interest">Special cases of common interest</h2></div>

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

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</math></span><img src="./26e23dc2d90da82d440a741a9793053c35c04d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.552ex; height:2.843ex;" alt="{\displaystyle (Y,{\cal {T}})}" loading="lazy"></span> 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 \mathbb {C} }">

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

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</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.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 {C} }" loading="lazy"></span> equipped with its usual topology. Then the essential range of <i>f</i> is given by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon \in \mathbb {R} _{>0}:0<\mu \{x\in X:|f(x)-z|<\varepsilon \}\right\}.}</annotation>

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</math></span><img src="./dbc8b058905c097875d480bf959e36b846d400f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.743ex; height:2.843ex;" alt="{\displaystyle \operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon \in \mathbb {R} _{>0}:0<\mu \{x\in X:|f(x)-z|<\varepsilon \}\right\}.}" loading="lazy"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Definition 4.36">: Definition 4.36 </span></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: cf. Exercise 6.11">: cf. Exercise 6.11 </span></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Exercise 3.19">: Exercise 3.19 </span></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Definition 2.61">: Definition 2.61 </span></sup></dd></dl>

<p>In other words: The essential range of a complex-valued function is the set of all complex numbers <i>z</i> such that the inverse image of each ε-neighbourhood of <i>z</i> under <i>f</i> has positive measure.

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<div class="mw-heading mw-heading3"><h3 id="(Y,T)_is_discrete">(<i>Y</i>,<i>T</i>) is discrete</h3></div>

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

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</math></span><img src="./26e23dc2d90da82d440a741a9793053c35c04d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.552ex; height:2.843ex;" alt="{\displaystyle (Y,{\cal {T}})}" loading="lazy"></span> is <a href="Discrete_space" title="Discrete space">discrete</a>, i.e., <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 {\cal {T}}={\cal {P}}(Y)}">

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</math></span><img src="./68d5dde3c7b8cdf4367a34ee4dc9a9dd1af5e3dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.321ex; height:2.843ex;" alt="{\displaystyle {\cal {T}}={\cal {P}}(Y)}" loading="lazy"></span> is the <a href="Power_set" title="Power set">power set</a> 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 Y,}">

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</math></span><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"></span> i.e., the <a href="Discrete_space#Definition" title="Discrete space">discrete topology</a> on <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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</math></span><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"></span> Then the essential range of <i>f</i> is the set of values <i>y</i> in <i>Y</i> with strictly positive <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_{*}\mu }">

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<annotation encoding="application/x-tex">{\displaystyle f_{*}\mu }</annotation>

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</math></span><img src="./c84eeadae4cbdb2162050c53d5a759f92fc2ae04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.595ex; height:2.676ex;" alt="{\displaystyle f_{*}\mu }" loading="lazy"></span>-measure:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.}</annotation>

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</math></span><img src="./e01d6b137741ac1ae68de4458bcecf58aceecb12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.416ex; height:2.843ex;" alt="{\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.}" loading="lazy"></span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Example 1.1.29">: Example 1.1.29 </span></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl>

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

<ul><li>The essential range of a measurable function, being the <a href="Support_(measure_theory)" title="Support (measure theory)">support of a measure</a>, is always closed.</li>

<li>The essential range ess.im(f) of a measurable function is always a 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 {\overline {\operatorname {im} (f)}}}">

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

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</math></span><img src="./93bad02e5aba4965f8109037d96c214714eea17a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.786ex; height:3.676ex;" alt="{\displaystyle {\overline {\operatorname {im} (f)}}}" loading="lazy"></span>.</li>

<li>The essential image cannot be used to distinguish functions that are almost everywhere equal: 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=g}">

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

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</math></span><img src="./795e79b6da5372a37ba3a36db68e43806232aac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.493ex; height:2.509ex;" alt="{\displaystyle f=g}" loading="lazy"></span> holds <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 \mu }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span>-<a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>, then <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 \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}</annotation>

</semantics>

</math></span><img src="./27d8de69e2d73581ef688494eabdf3565b47eff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.075ex; height:2.843ex;" alt="{\displaystyle \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}" loading="lazy"></span>.</li>

<li>These two facts characterise the essential image: It is the biggest set contained in the closures 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 \operatorname {im} (g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {im} (g)}</annotation>

</semantics>

</math></span><img src="./2e2f3153c2453e1ff18b710d2837fdff16363959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.508ex; height:2.843ex;" alt="{\displaystyle \operatorname {im} (g)}" loading="lazy"></span> for all g that are a.e. equal to f:</li></ul>

</mrow>

</mrow>

</munder>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}}</annotation>

</semantics>

</math></span><img src="./fbfd68642ad392cecb3675fa358e0b540e998b30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.504ex; height:6.176ex;" alt="{\displaystyle \operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}}" loading="lazy"></span>.</dd></dl></dd></dl>

<ul><li>The essential range satisfies <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 \forall A\subseteq X:f(A)\cap \operatorname {ess.im} (f)=\emptyset \implies \mu (A)=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall A\subseteq X:f(A)\cap \operatorname {ess.im} (f)=\emptyset \implies \mu (A)=0}</annotation>

</semantics>

</math></span><img src="./22f5e5e7095b8afc34f9823ffc906bcc0ff7cf72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.896ex; height:2.843ex;" alt="{\displaystyle \forall A\subseteq X:f(A)\cap \operatorname {ess.im} (f)=\emptyset \implies \mu (A)=0}" loading="lazy"></span>.</li>

<li>This fact characterises the essential image: It is the <i>smallest</i> closed 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 \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.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 {C} }" loading="lazy"></span> with this property.</li>

<li>The <a href="Essential_supremum" class="mw-redirect" title="Essential supremum">essential supremum</a> of a real valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function is essentially bounded if and only if its essential range is bounded.</li>

<li>The essential range of an essentially bounded function f is equal to the <a href="Spectrum_(functional_analysis)#Spectrum_of_a_unital_Banach_algebra" title="Spectrum (functional analysis)">spectrum</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 (f)}">

</mstyle>

</mrow>

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

</semantics>

</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> where f is considered as an element of the <a href="C*-algebra" title="C*-algebra">C*-algebra</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 L^{\infty }(\mu )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5b7867eb72dc22e91568af1af857fd364f42458c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.669ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )}" loading="lazy"></span>.</li></ul>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> is the zero measure, then the essential image of all measurable functions is empty.</li>

<li>This also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.</li>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4538114b7e232be1bd1d0d774e97f5c43236518a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.975ex; height:2.509ex;" alt="{\displaystyle X\subseteq \mathbb {R} ^{n}}" loading="lazy"></span> is open, <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> continuous 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 \mu }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>, then <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 \operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}}">

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}}</annotation>

</semantics>

</math></span><img src="./2d3231bbf5bef63645ff82739f096ef642a8e07b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.453ex; height:3.676ex;" alt="{\displaystyle \operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}}" loading="lazy"></span> holds. This holds more generally for all <a href="Borel_measure" title="Borel measure">Borel measures</a> that assign non-zero measure to every non-empty open set.</li></ul>

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

<p>The notion of essential range can be extended to the case 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:X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</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>, 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 <a href="Separable_space" title="Separable space">separable</a> <a href="Metric_space" title="Metric space">metric space</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 X}">

</mstyle>

</mrow>

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

</semantics>

</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}">

</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> are <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a> of the same dimension, 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\in }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6a56fbab489227ab3e129a458c966d1ef01ab938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.474ex; height:2.509ex;" alt="{\displaystyle f\in }" loading="lazy"></span> <a href="Bounded_mean_oscillation#The_space_VMO" title="Bounded mean oscillation">VMO</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 (X,Y)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./41f29b9537685f499713112d6802e811cbf51bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}" loading="lazy"></span> and 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 \operatorname {ess.im} (f)\neq Y}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ess.im} (f)\neq Y}</annotation>

</semantics>

</math></span><img src="./ff61f3cb1ad5ff07cf881b9ffc49e12ac03e6fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.441ex; height:2.843ex;" alt="{\displaystyle \operatorname {ess.im} (f)\neq Y}" loading="lazy"></span>, then <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 \deg f=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \deg f=0}</annotation>

</semantics>

</math></span><img src="./21847e845e35d439224207649980da777c5ac4f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.414ex; height:2.509ex;" alt="{\displaystyle \deg f=0}" loading="lazy"></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Essential_supremum_and_essential_infimum" class="mw-redirect" title="Essential supremum and essential infimum">Essential supremum and essential infimum</a></li>

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

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

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

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFZimmer1990" class="citation book cs1"><a href="Robert_Zimmer" title="Robert Zimmer">Zimmer, Robert J.</a> (1990). <i>Essential Results of Functional Analysis</i>. University of Chicago Press. p. 2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-226-98337-4</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFKuksinShirikyan2012" class="citation book cs1"><a href="Sergei_B._Kuksin" title="Sergei B. Kuksin">Kuksin, Sergei</a>; Shirikyan, Armen (2012). <i>Mathematics of Two-Dimensional Turbulence</i>. Cambridge University Press. p. 292. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-02282-9</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFKon1985" class="citation book cs1">Kon, Mark A. (1985). <i>Probability Distributions in Quantum Statistical Mechanics</i>. Springer. pp. 74, 84. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-15690-9</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFDriver2012" class="citation book cs1">Driver, Bruce (May 7, 2012). <a rel="nofollow" class="external text" href="https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf"><i>Analysis Tools with Examples</i></a> <span class="cs1-format">(PDF)</span>. p. 327.</cite> Cf. Exercise 30.5.1.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFSegalKunze1978" class="citation book cs1"><a href="Irving_Segal" title="Irving Segal">Segal, Irving E.</a>; <a href="Ray_Kunze" title="Ray Kunze">Kunze, Ray A.</a> (1978). <i>Integrals and Operators</i> (2nd revised and enlarged ed.). Springer. p. 106. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-08323-5</bdi>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFBogachevSmolyanov2020" class="citation book cs1">Bogachev, Vladimir I.; Smolyanov, Oleg G. (2020). <i>Real and Functional Analysis</i>. Moscow Lectures. Springer. p. 283. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-38219-3</bdi>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2522-0314">2522-0314</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeaver2013" class="citation book cs1">Weaver, Nik (2013). <i>Measure Theory and Functional Analysis</i>. World Scientific. p. 142. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-4508-56-8</bdi>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFBhatia2009" class="citation book cs1"><a href="Rajendra_Bhatia" title="Rajendra Bhatia">Bhatia, Rajendra</a> (2009). <i>Notes on Functional Analysis</i>. Hindustan Book Agency. p. 149. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-85931-89-0</bdi>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFFolland1999" class="citation book cs1"><a href="Gerald_Folland" title="Gerald Folland">Folland, Gerald B.</a> (1999). <i>Real Analysis: Modern Techniques and Their Applications</i>. Wiley. p. 187. <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-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFRudin1987" class="citation book cs1">Rudin, Walter (1987). <i>Real and complex analysis</i> (3rd ed.). New York: McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-054234-1</bdi>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFDouglas1998" class="citation book cs1">Douglas, Ronald G. (1998). <i>Banach algebra techniques in operator theory</i> (2nd ed.). New York Berlin Heidelberg: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98377-5</bdi>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Cf. <cite id="CITEREFTao2012" class="citation book cs1"><a href="Terence_Tao" title="Terence Tao">Tao, Terence</a> (2012). <i>Topics in Random Matrix Theory</i>. American Mathematical Society. p. 29. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-7430-1</bdi>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Cf. <cite id="CITEREFFreedman1971" class="citation book cs1"><a href="David_A._Freedman" title="David A. Freedman">Freedman, David</a> (1971). <i>Markov Chains</i>. Holden-Day. p. 1.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Cf. <cite id="CITEREFChung1967" class="citation book cs1"><a href="Chung_Kai-lai" title="Chung Kai-lai">Chung, Kai Lai</a> (1967). <i>Markov Chains with Stationary Transition Probabilities</i>. Springer. p. 135.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrezisNirenberg1995" class="citation journal cs1">Brezis, Haïm; Nirenberg, Louis (September 1995). "Degree theory and BMO. Part I: Compact manifolds without boundaries". <i>Selecta Mathematica</i>. <b>1</b> (2): <span class="nowrap">197–</span>263. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01671566">10.1007/BF01671566</a>.</cite></span>

</li>

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

<ul><li><cite id="CITEREFWalter_Rudin1974" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Walter Rudin</a> (1974). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/realcomplexanaly00rudi_0"><i>Real and Complex Analysis</i></a></span> (2nd ed.). <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054234-1</bdi>.</cite></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>/<a href="Measurable_function" title="Measurable function">function</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>

<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="Measurable_function" title="Measurable function">Measurable function</a>

<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><div><div class="zim-footer">

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