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<title>Random compact set</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Random compact set</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>, a <b>random compact set</b> is essentially a <a href="Compact_space" title="Compact space">compact set</a>-valued <a href="Random_variable" title="Random variable">random variable</a>. Random compact sets are useful in the study of attractors for <a href="Random_dynamical_system" title="Random dynamical system">random dynamical systems</a>.

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

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

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</math></span><img src="./d78e6f2ddf5baee227ee2a9f164726ba0c23c263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (M,d)}" loading="lazy"></span> be a <a href="Complete_space" class="mw-redirect" title="Complete space">complete</a> <a href="Separable_space" title="Separable space">separable</a> <a href="Metric_space" title="Metric space">metric space</a>. 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 {\mathcal {K}}}">

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

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</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span> denote the set of all compact subsets 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 M}">

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

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>. The Hausdorff metric <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 h}">

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

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</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> 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 {\mathcal {K}}}">

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

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</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span> is defined by

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<annotation encoding="application/x-tex">{\displaystyle h(K_{1},K_{2}):=\max \left\{\sup _{a\in K_{1}}\inf _{b\in K_{2}}d(a,b),\sup _{b\in K_{2}}\inf _{a\in K_{1}}d(a,b)\right\}.}</annotation>

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</math></span><img src="./31cb6e7540ed1218a3cb7557b1273a2365fd1bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.072ex; height:7.509ex;" alt="{\displaystyle h(K_{1},K_{2}):=\max \left\{\sup _{a\in K_{1}}\inf _{b\in K_{2}}d(a,b),\sup _{b\in K_{2}}\inf _{a\in K_{1}}d(a,b)\right\}.}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {K}},h)}</annotation>

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</math></span><img src="./77799a9013e23eb7fa77f49eadd7b1272d8e89a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.953ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {K}},h)}" loading="lazy"></span> is also а complete separable metric space. The corresponding open subsets generate a <a href="Sigma_algebra" class="mw-redirect" title="Sigma algebra">σ-algebra</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 {\mathcal {K}}}">

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

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</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span>, the <a href="Borel_sigma_algebra" class="mw-redirect" title="Borel sigma algebra">Borel sigma 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 {\mathcal {B}}({\mathcal {K}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}({\mathcal {K}})}</annotation>

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</math></span><img src="./f380c186f6df6283689998013e31c15306fd62be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.124ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}({\mathcal {K}})}" 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 {\mathcal {K}}}">

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

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</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span>.

</p><p>A <b>random compact set</b> is а <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 K}">

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

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</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" 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}" loading="lazy"></span> from а <a href="Probability_space" title="Probability space">probability space</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 (\Omega ,{\mathcal {F}},\mathbb {P} )}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}</annotation>

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</math></span><img src="./6bb8743f7565082ed1a9ee0490d9d71be82eafaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}" loading="lazy"></span> into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}</annotation>

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</math></span><img src="./927d1a33c8d33e124a53c25b1397cd0c19df91bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.738ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}" loading="lazy"></span>.

</p><p>Put another way, a random compact set is a measurable function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\colon \Omega \to 2^{M}}">

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<annotation encoding="application/x-tex">{\displaystyle K\colon \Omega \to 2^{M}}</annotation>

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</math></span><img src="./b30b1c57552c73f5f747e054dd35eb069608bf6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.514ex; height:2.676ex;" alt="{\displaystyle K\colon \Omega \to 2^{M}}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\omega )}">

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

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</math></span><img src="./21bef70b39ac777e9fe32525939280902d41b56d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.321ex; height:2.843ex;" alt="{\displaystyle K(\omega )}" loading="lazy"></span> is <a href="Almost_surely" title="Almost surely">almost surely</a> compact and

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<annotation encoding="application/x-tex">{\displaystyle \omega \mapsto \inf _{b\in K(\omega )}d(x,b)}</annotation>

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</math></span><img src="./658f5b388674d664aaecea6fc8d58a884c225127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.398ex; height:4.509ex;" alt="{\displaystyle \omega \mapsto \inf _{b\in K(\omega )}d(x,b)}" loading="lazy"></span></dd></dl>

<p>is a measurable function for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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

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</math></span><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span>.

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

<p>Random compact sets in this sense are also random closed sets as in <a href="Georges_Matheron" title="Georges Matheron">Matheron</a> (1975). Consequently, under the additional assumption that the carrier space is locally compact, their distribution is given by the probabilities

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (X\cap K=\emptyset )}</annotation>

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</math></span><img src="./b88bac02642d75ea57412ce1a8c2571a29b67b94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.119ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (X\cap K=\emptyset )}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\in {\mathcal {K}}.}">

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

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</math></span><img src="./8983ece62573986aec506ed44fb9480f75620d8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.324ex; height:2.176ex;" alt="{\displaystyle K\in {\mathcal {K}}.}" loading="lazy"></span></dd></dl>

<p>(The distribution of а random compact convex set is also given by the system of all inclusion probabilities <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 {P} (X\subset K).}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (X\subset K).}</annotation>

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</math></span><img src="./5b90bdad07d1f05940dc4c54fce43e66815763b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.021ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (X\subset K).}" loading="lazy"></span>)

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

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

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</math></span><img src="./a7057f0242dfbaa91fbe48b7647f96382c4576a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.819ex; height:2.843ex;" alt="{\displaystyle K=\{x\}}" loading="lazy"></span>, the probability <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 {P} (x\in X)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (x\in X)}</annotation>

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</math></span><img src="./ea1a1b5416304548020131fc64fd085e8e92ff6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.38ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (x\in X)}" loading="lazy"></span> is obtained, which satisfies

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (x\in X)=1-\mathbb {P} (x\not \in X).}</annotation>

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</math></span><img src="./d6a5facb7e16e525034be1565569c0bf86c9e696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.508ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (x\in X)=1-\mathbb {P} (x\not \in X).}" loading="lazy"></span></dd></dl>

<p>Thus the <b>covering function</b> <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 p_{X}}">

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

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</math></span><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy"></span> is given by

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<annotation encoding="application/x-tex">{\displaystyle p_{X}(x)=\mathbb {P} (x\in X)}</annotation>

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</math></span><img src="./9d07b75b5ff1ed7ba0149d5bd821a63632afe8e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:18.509ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)=\mathbb {P} (x\in X)}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M.}">

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

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</math></span><img src="./127645d1eb7572c38f34e5b855f5d789404f6d93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.259ex; height:2.176ex;" alt="{\displaystyle x\in M.}" loading="lazy"></span></dd></dl>

<p>Of course, <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 p_{X}}">

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

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</math></span><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy"></span> can also be interpreted as the mean of the indicator function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{X}}">

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

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</math></span><img src="./8d50568f98917dc6576fe281b18ee4e111038b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.969ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{X}}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle p_{X}(x)=\mathbb {E} \mathbf {1} _{X}(x).}</annotation>

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</math></span><img src="./b4a0eab12fd946935384d3c41d4212b2a27d0f1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.434ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)=\mathbb {E} \mathbf {1} _{X}(x).}" loading="lazy"></span></dd></dl>

<p>The covering function takes values between <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 0}">

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

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</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" 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 1}">

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

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>. The set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{X}}">

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

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</math></span><img src="./f75cba15dfdf57b5ffac5911cb209fe0e0e45a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.63ex; height:2.509ex;" alt="{\displaystyle b_{X}}" loading="lazy"></span> of all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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

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</math></span><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span> with <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 p_{X}(x)>0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{X}(x)>0}</annotation>

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</math></span><img src="./d6aa2c6e3dd92b338fad824ba6ed560f279032f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.291ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)>0}" loading="lazy"></span> is called the <b>support</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 X}">

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

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. The set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{X}}">

<msub>

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

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

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</math></span><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span> with <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 p_{X}(x)=1}">

<msub>

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

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</math></span><img src="./5254614a0e3493990e6c03dc1087b117c66ab1b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.291ex; height:2.843ex;" alt="{\displaystyle p_{X}(x)=1}" loading="lazy"></span> is called the <b>kernel</b>, the set of <b>fixed points</b>, or <b>essential minimum</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(X)}">

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

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</math></span><img src="./54526d2769dc9af6cbd61c1747c4da8b3491be7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.873ex; height:2.843ex;" alt="{\displaystyle e(X)}" loading="lazy"></span>. 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_{1},X_{2},\ldots }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots }</annotation>

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</math></span><img src="./869cccabc3ea7b90d40e40158d740fc88e07889a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots }" loading="lazy"></span>, is а sequence of <a href="I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> random compact sets, then almost surely

</p>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigcap _{i=1}^{\infty }X_{i}=e(X)}</annotation>

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</math></span><img src="./47ec9449e8c7364bb4b32ed29bb6cb8db963f1ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.75ex; height:6.843ex;" alt="{\displaystyle \bigcap _{i=1}^{\infty }X_{i}=e(X)}" loading="lazy"></span></dd></dl>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigcap _{i=1}^{\infty }X_{i}}</annotation>

</semantics>

</math></span><img src="./5701a70b0e894e90ffaafee5695a0fb444a8d053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.779ex; height:6.843ex;" alt="{\displaystyle \bigcap _{i=1}^{\infty }X_{i}}" loading="lazy"></span> converges almost surely to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(X).}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./57802d839fd9fa61759091a233f6ac105707f172.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle e(X).}" loading="lazy"></span>

</p>

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

<ul><li>Matheron, G. (1975) <i>Random Sets and Integral Geometry</i>. J.Wiley & Sons, New York. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-57621-2</bdi></li>

<li>Molchanov, I. (2005) <i>The Theory of Random Sets</i>. Springer, New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-85233-892-X</bdi></li>

<li>Stoyan D., and H.Stoyan (1994) <i>Fractals, Random Shapes and Point Fields</i>. John Wiley & Sons, Chichester, New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-93757-6</bdi></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Measure_theory138" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<ul><li><a href="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: </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">

This article is issued from <a class="external text" title="Last edited on 2023-06-17" href="https://en.wikipedia.org/wiki/?title=Random_compact_set&oldid=1160630683">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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