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searchaux" style="display:none">Relates tightness of measures to relative compactness in the space of probability measures</div> <p>In <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> <b>Prokhorov's theorem</b> relates <a href="/wiki/Tightness_of_measures" title="Tightness of measures">tightness of measures</a> to relative <a href="/wiki/Compactness" class="mw-redirect" title="Compactness">compactness</a> (and hence <a href="/wiki/Weak_convergence_of_measures" class="mw-redirect" title="Weak convergence of measures">weak convergence</a>) in the space of <a href="/wiki/Probability_measure" title="Probability measure">probability measures</a>. It is credited to the <a href="/wiki/Soviet_Union" title="Soviet Union">Soviet</a> mathematician <a href="/wiki/Yuri_Vasilevich_Prokhorov" class="mw-redirect" title="Yuri Vasilevich Prokhorov">Yuri Vasilyevich Prokhorov</a>, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prokhorov%27s_theorem&action=edit&section=1" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e108e14bd8889585a2c0a0beae69006bd69f7033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.544ex; height:2.843ex;" alt="{\displaystyle (S,\rho )}"></span> be a <a href="/wiki/Separable_space" title="Separable space">separable</a> <a href="/wiki/Metric_space" title="Metric space">metric space</a>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b51bfe21fd15a7b5531fd9635a87a3863be6025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(S)}"></span> denote the collection of all probability measures defined on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> (with its <a href="/wiki/Borel_sigma_algebra" class="mw-redirect" title="Borel sigma algebra">Borel σ-algebra</a>). </p><p><b>Theorem.</b> </p> <ol><li>A collection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbdfdf84109a92ad3afd4caee807455f1eef141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.177ex; height:2.843ex;" alt="{\displaystyle K\subset {\mathcal {P}}(S)}"></span> of probability measures is <a href="/wiki/Tightness_of_measures" title="Tightness of measures">tight</a> if and only if the closure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is <a href="/wiki/Sequentially_compact" class="mw-redirect" title="Sequentially compact">sequentially compact</a> in the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b51bfe21fd15a7b5531fd9635a87a3863be6025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(S)}"></span> equipped with the <a href="/wiki/Topology" title="Topology">topology</a> of <a href="/wiki/Weak_convergence_of_measures" class="mw-redirect" title="Weak convergence of measures">weak convergence</a>.</li> <li>The space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b51bfe21fd15a7b5531fd9635a87a3863be6025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(S)}"></span> with the topology of weak convergence is <a href="/wiki/Metrizable" class="mw-redirect" title="Metrizable">metrizable</a>.</li> <li>Suppose that in addition, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e108e14bd8889585a2c0a0beae69006bd69f7033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.544ex; height:2.843ex;" alt="{\displaystyle (S,\rho )}"></span> is a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a> (so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e108e14bd8889585a2c0a0beae69006bd69f7033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.544ex; height:2.843ex;" alt="{\displaystyle (S,\rho )}"></span> is a <a href="/wiki/Polish_space" title="Polish space">Polish space</a>). There is a complete metric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4740381c16ea98c4132510daa642e93c1e42c049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{0}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b51bfe21fd15a7b5531fd9635a87a3863be6025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(S)}"></span> equivalent to the topology of weak convergence; moreover, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbdfdf84109a92ad3afd4caee807455f1eef141" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.177ex; height:2.843ex;" alt="{\displaystyle K\subset {\mathcal {P}}(S)}"></span> is tight if and only if the <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {P}}(S),d_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {P}}(S),d_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a3c2e8ffcbc3c8595bfca9e9a70db789ccece2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.119ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {P}}(S),d_{0})}"></span> is compact.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Corollaries">Corollaries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prokhorov%27s_theorem&action=edit&section=2" title="Edit section: Corollaries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For Euclidean spaces we have that: </p> <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c3c92ca8fa9fd9429dde17e2a4dc21f62104fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.429ex; height:2.843ex;" alt="{\displaystyle (\mu _{n})}"></span> is a tight <a href="/wiki/Sequence" title="Sequence">sequence</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a76f1e675f8aed6c2a7261722dd75c415bb4a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.866ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}"></span> (the collection of probability measures on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>), then there exist a <a href="/wiki/Subsequence" title="Subsequence">subsequence</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{n_{k}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{n_{k}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd601929dc808c5cf63e72a70e9867677922c077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.289ex; height:3.009ex;" alt="{\displaystyle (\mu _{n_{k}})}"></span> and a probability measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733b3d92f331bcfde4a4ae08f9c8f98ed9f06c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.108ex; height:2.843ex;" alt="{\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5f18cf19c9aa13145c64836bfdec5033ff1510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.48ex; height:2.343ex;" alt="{\displaystyle \mu _{n_{k}}}"></span> converges weakly to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c3c92ca8fa9fd9429dde17e2a4dc21f62104fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.429ex; height:2.843ex;" alt="{\displaystyle (\mu _{n})}"></span> is a tight sequence in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a76f1e675f8aed6c2a7261722dd75c415bb4a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.866ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}"></span> such that every weakly convergent subsequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{n_{k}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{n_{k}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd601929dc808c5cf63e72a70e9867677922c077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.289ex; height:3.009ex;" alt="{\displaystyle (\mu _{n_{k}})}"></span> has the same limit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733b3d92f331bcfde4a4ae08f9c8f98ed9f06c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.108ex; height:2.843ex;" alt="{\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}"></span>, then the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c3c92ca8fa9fd9429dde17e2a4dc21f62104fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.429ex; height:2.843ex;" alt="{\displaystyle (\mu _{n})}"></span> converges weakly to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Extension">Extension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prokhorov%27s_theorem&action=edit&section=3" title="Edit section: Extension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prokhorov's theorem can be extended to consider <a href="/wiki/Complex_measure" title="Complex measure">complex measures</a> or finite <a href="/wiki/Signed_measure" title="Signed measure">signed measures</a>. </p><p><b>Theorem:</b> Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e108e14bd8889585a2c0a0beae69006bd69f7033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.544ex; height:2.843ex;" alt="{\displaystyle (S,\rho )}"></span> is a complete separable metric space and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e3db6cc2028a183af948212ed2551d25c954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }"></span> is a family of Borel complex measures on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. The following statements are equivalent: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e3db6cc2028a183af948212ed2551d25c954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }"></span> is sequentially precompact; that is, every sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mu _{n}\}\subset \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⊂</mo> <mi mathvariant="normal">Π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mu _{n}\}\subset \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0a94849df9a1c959308c8275490d5cdac71f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.787ex; height:2.843ex;" alt="{\displaystyle \{\mu _{n}\}\subset \Pi }"></span> has a weakly convergent subsequence.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e3db6cc2028a183af948212ed2551d25c954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }"></span> is tight and uniformly bounded in <a href="/wiki/Total_variation#Total_variation_in_measure_theory" title="Total variation">total variation norm</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Comments">Comments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prokhorov%27s_theorem&action=edit&section=4" title="Edit section: Comments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since Prokhorov's theorem expresses tightness in terms of compactness, the <a href="/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem" title="Arzelà–Ascoli theorem">Arzelà–Ascoli theorem</a> is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the <a href="/wiki/Modulus_of_continuity" title="Modulus of continuity">modulus of continuity</a> or an appropriate analogue—see <a href="/wiki/Classical_Wiener_space#Tightness_in_classical_Wiener_space" title="Classical Wiener space">tightness in classical Wiener space</a> and <a href="/wiki/C%C3%A0dl%C3%A0g#Tightness_in_Skorokhod_space" title="Càdlàg">tightness in Skorokhod space</a>. </p><p>There are several deep and non-trivial extensions to Prokhorov's theorem. However, those results do not overshadow the importance and the relevance to applications of the original result. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prokhorov%27s_theorem&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a> – certain metric on space of finite measures<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Sazonov%27s_theorem" title="Sazonov's theorem">Sazonov's theorem</a></li> <li><a href="/wiki/Tightness_of_measures" title="Tightness of measures">Tightness of measures</a> – Concept in measure theory</li> <li><a href="/wiki/Weak_convergence_of_measures" class="mw-redirect" title="Weak convergence of measures">Weak convergence of measures</a> – Mathematical concept<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prokhorov%27s_theorem&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBillingsley1999" class="citation book cs1">Billingsley, Patrick (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/convergenceofpro0000bill"><i>Convergence of Probability Measures</i></a></span>. New York, NY: John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-19745-9" title="Special:BookSources/0-471-19745-9"><bdi>0-471-19745-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convergence+of+Probability+Measures&rft.place=New+York%2C+NY&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1999&rft.isbn=0-471-19745-9&rft.aulast=Billingsley&rft.aufirst=Patrick&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconvergenceofpro0000bill&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProkhorov%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBogachev2006" class="citation book cs1">Bogachev, Vladimir (2006). <i>Measure Theory Vol 1 and 2</i>. 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"Convergence of random processes and limit theorems in probability theory". <i>Theory of Probability & Its Applications</i>. <b>1</b> (2): 157–214. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1101016">10.1137/1101016</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+of+Probability+%26+Its+Applications&rft.atitle=Convergence+of+random+processes+and+limit+theorems+in+probability+theory&rft.volume=1&rft.issue=2&rft.pages=157-214&rft.date=1956&rft_id=info%3Adoi%2F10.1137%2F1101016&rft.aulast=Prokhorov&rft.aufirst=Yuri+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProkhorov%27s+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudley1989" class="citation book cs1">Dudley, Richard. M. 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Chapman & Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-05161-3" title="Special:BookSources/0-412-05161-3"><bdi>0-412-05161-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+analysis+and+Probability&rft.pub=Chapman+%26+Hall&rft.date=1989&rft.isbn=0-412-05161-3&rft.aulast=Dudley&rft.aufirst=Richard.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProkhorov%27s+theorem" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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"}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Measure_theory" title="Template:Measure theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Measure_theory" title="Template talk:Measure theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Measure_theory" title="Special:EditPage/Template:Measure theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Measure_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity</a> <a href="/wiki/Absolute_continuity_(measure_theory)" class="mw-redirect" title="Absolute continuity (measure theory)">of measures</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a></li> <li><a href="/wiki/Measure_space" title="Measure space">Measure space</a> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li></ul></li> <li><a href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/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="/wiki/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Atom_(measure_theory)" title="Atom (measure theory)">Atom</a></li> <li><a href="/wiki/Baire_set" title="Baire set">Baire set</a></li> <li><a href="/wiki/Borel_set" title="Borel set">Borel set</a> <ul><li><a href="/wiki/Borel_equivalence_relation" title="Borel equivalence relation">equivalence relation</a></li></ul></li> <li><a href="/wiki/Standard_Borel_space" title="Standard Borel space">Borel space</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_criterion" title="Carathéodory's criterion">Carathéodory's criterion</a></li> <li><a href="/wiki/Cylindrical_%CF%83-algebra" title="Cylindrical σ-algebra">Cylindrical σ-algebra</a> <ul><li><a href="/wiki/Cylinder_set" title="Cylinder set">Cylinder set</a></li></ul></li> <li><a href="/wiki/Dynkin_system" title="Dynkin system">𝜆-system</a></li> <li><a href="/wiki/Essential_range" title="Essential range">Essential range</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">infimum/supremum</a></li></ul></li> <li><a href="/wiki/Locally_measurable_set" class="mw-redirect" title="Locally measurable set">Locally measurable</a></li> <li><a href="/wiki/Pi-system" title="Pi-system"><span class="texhtml mvar" style="font-style:italic;">π</span>-system</a></li> <li><a href="/wiki/%CE%A3-algebra" title="Σ-algebra">σ-algebra</a></li> <li><a href="/wiki/Non-measurable_set" title="Non-measurable set">Non-measurable set</a> <ul><li><a href="/wiki/Vitali_set" title="Vitali set">Vitali set</a></li></ul></li> <li><a href="/wiki/Null_set" title="Null set">Null set</a></li> <li><a href="/wiki/Support_(measure_theory)" title="Support (measure theory)">Support</a></li> <li><a href="/wiki/Transverse_measure" title="Transverse measure">Transverse measure</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Atomic_measure" class="mw-redirect" title="Atomic measure">Atomic</a></li> <li><a href="/wiki/Baire_measure" title="Baire measure">Baire</a></li> <li><a href="/wiki/Banach_measure" title="Banach measure">Banach</a></li> <li><a href="/wiki/Besov_measure" title="Besov measure">Besov</a></li> <li><a href="/wiki/Borel_measure" title="Borel measure">Borel</a></li> <li><a href="/wiki/Brown_measure" title="Brown measure">Brown</a></li> <li><a href="/wiki/Complex_measure" title="Complex measure">Complex</a></li> <li><a href="/wiki/Complete_measure" title="Complete measure">Complete</a></li> <li><a href="/wiki/Content_(measure_theory)" title="Content (measure theory)">Content</a></li> <li>(<a href="/wiki/Logarithmically_concave_measure" title="Logarithmically concave measure">Logarithmically</a>) <a href="/wiki/Convex_measure" title="Convex measure">Convex</a></li> <li><a href="/wiki/Decomposable_measure" title="Decomposable measure">Decomposable</a></li> <li><a href="/wiki/Discrete_measure" title="Discrete measure">Discrete</a></li> <li><a href="/wiki/Equivalence_(measure_theory)" title="Equivalence (measure theory)">Equivalent</a></li> <li><a href="/wiki/Finite_measure" title="Finite measure">Finite</a></li> <li><a href="/wiki/Inner_measure" title="Inner measure">Inner</a></li> <li>(<a href="/wiki/Quasi-invariant_measure" title="Quasi-invariant measure">Quasi-</a>) <a href="/wiki/Invariant_measure" title="Invariant measure">Invariant</a></li> <li><a href="/wiki/Locally_finite_measure" title="Locally finite measure">Locally finite</a></li> <li><a href="/wiki/Maximising_measure" title="Maximising measure">Maximising</a></li> <li><a href="/wiki/Metric_outer_measure" title="Metric outer measure">Metric outer</a></li> <li><a href="/wiki/Outer_measure" title="Outer measure">Outer</a></li> <li><a href="/wiki/Perfect_measure" title="Perfect measure">Perfect</a></li> <li><a href="/wiki/Pre-measure" title="Pre-measure">Pre-measure</a></li> <li>(<a href="/wiki/Sub-probability_measure" title="Sub-probability measure">Sub-</a>) <a href="/wiki/Probability_measure" title="Probability measure">Probability</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Radon_measure" title="Radon measure">Radon</a></li> <li><a href="/wiki/Random_measure" title="Random measure">Random</a></li> <li><a href="/wiki/Regular_measure" title="Regular measure">Regular</a> <ul><li><a href="/wiki/Borel_regular_measure" title="Borel regular measure">Borel regular</a></li> <li><a href="/wiki/Inner_regular_measure" class="mw-redirect" title="Inner regular measure">Inner regular</a></li> <li><a href="/wiki/Outer_regular_measure" class="mw-redirect" title="Outer regular measure">Outer regular</a></li></ul></li> <li><a href="/wiki/Saturated_measure" title="Saturated measure">Saturated</a></li> <li><a href="/wiki/Set_function" title="Set function">Set function</a></li> <li><a href="/wiki/%CE%A3-finite_measure" title="Σ-finite measure">σ-finite</a></li> <li><a href="/wiki/S-finite_measure" title="S-finite measure">s-finite</a></li> <li><a href="/wiki/Signed_measure" title="Signed measure">Signed</a></li> <li><a href="/wiki/Singular_measure" title="Singular measure">Singular</a></li> <li><a href="/wiki/Spectral_measure" class="mw-redirect" title="Spectral measure">Spectral</a></li> <li><a href="/wiki/Strictly_positive_measure" title="Strictly positive measure">Strictly positive</a></li> <li><a href="/wiki/Tightness_of_measures" title="Tightness of measures">Tight</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Measures_(measure_theory)" title="Category:Measures (measure theory)">Particular measures</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="/wiki/Counting_measure" title="Counting measure">Counting</a></li> <li><a href="/wiki/Dirac_measure" title="Dirac measure">Dirac</a></li> <li><a href="/wiki/Euler_measure" title="Euler measure">Euler</a></li> <li><a href="/wiki/Gaussian_measure" title="Gaussian measure">Gaussian</a></li> <li><a href="/wiki/Haar_measure" title="Haar measure">Haar</a></li> <li><a href="/wiki/Harmonic_measure" title="Harmonic measure">Harmonic</a></li> <li><a href="/wiki/Hausdorff_measure" title="Hausdorff measure">Hausdorff</a></li> <li><a href="/wiki/Intensity_measure" title="Intensity measure">Intensity</a></li> <li><a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Infinite-dimensional</a></li></ul></li> <li><a href="/wiki/Positive_real_numbers#Logarithmic_measure" title="Positive real numbers">Logarithmic</a></li> <li><a href="/wiki/Product_measure" title="Product measure">Product</a> <ul><li><a href="/wiki/Projection_(measure_theory)" title="Projection (measure theory)">Projections</a></li></ul></li> <li><a href="/wiki/Pushforward_measure" title="Pushforward measure">Pushforward</a></li> <li><a href="/wiki/Spherical_measure" title="Spherical measure">Spherical measure</a></li> <li><a href="/wiki/Tangent_measure" title="Tangent measure">Tangent</a></li> <li><a href="/wiki/Trivial_measure" title="Trivial measure">Trivial</a></li> <li><a href="/wiki/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="/wiki/Measurable_function" title="Measurable function">Measurable function</a> <ul><li><a href="/wiki/Bochner_measurable_function" title="Bochner measurable function">Bochner</a></li> <li><a href="/wiki/Strongly_measurable_function" title="Strongly measurable function">Strongly</a></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a></li></ul></li> <li>Convergence: <a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">almost everywhere</a></li> <li><a href="/wiki/Convergence_of_measures" title="Convergence of measures">of measures</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">in measure</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">of random variables</a> <ul><li><a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">in distribution</a></li> <li><a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">in probability</a></li></ul></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li> <li>Random: <a href="/wiki/Random_compact_set" title="Random compact set">compact set</a></li> <li><a href="/wiki/Random_element" title="Random element">element</a></li> <li><a href="/wiki/Random_measure" title="Random measure">measure</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">process</a></li> <li><a href="/wiki/Random_variable" title="Random variable">variable</a></li> <li><a href="/wiki/Multivariate_random_variable" title="Multivariate random variable">vector</a></li> <li><a href="/wiki/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%"><a href="/wiki/Category:Theorems_in_measure_theory" title="Category:Theorems in measure theory">Main results</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="/wiki/Carath%C3%A9odory%27s_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a></li> <li>Convergence theorems <ul><li><a href="/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">Dominated</a></li> <li><a href="/wiki/Monotone_convergence_theorem" title="Monotone convergence theorem">Monotone</a></li> <li><a href="/wiki/Vitali_convergence_theorem" title="Vitali convergence theorem">Vitali</a></li></ul></li> <li>Decomposition theorems <ul><li><a href="/wiki/Hahn_decomposition_theorem" title="Hahn decomposition theorem">Hahn</a></li> <li><a href="/wiki/Jordan_decomposition_theorem" class="mw-redirect" title="Jordan decomposition theorem">Jordan</a></li> <li><a href="/wiki/Maharam%27s_theorem" title="Maharam's theorem">Maharam's</a></li></ul></li> <li><a href="/wiki/Egorov%27s_theorem" title="Egorov's theorem">Egorov's</a></li> <li><a href="/wiki/Fatou%27s_lemma" title="Fatou's lemma">Fatou's lemma</a></li> <li><a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's</a> <ul><li><a href="/wiki/Fubini%E2%80%93Tonelli_theorem" class="mw-redirect" title="Fubini–Tonelli theorem">Fubini–Tonelli</a></li></ul></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a></li> <li><a href="/wiki/Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym</a></li> <li><a href="/wiki/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="/wiki/Disintegration_theorem" title="Disintegration theorem">Disintegration theorem</a> <ul><li><a href="/wiki/Lifting_theory" title="Lifting theory">Lifting theory</a></li></ul></li> <li><a href="/wiki/Lebesgue%27s_density_theorem" title="Lebesgue's density theorem">Lebesgue's density theorem</a></li> <li><a href="/wiki/Lebesgue_differentiation_theorem" title="Lebesgue differentiation theorem">Lebesgue differentiation theorem</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's theorem</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_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="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a></li> <li><a href="/wiki/Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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