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<title>Intensitätsmaß</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Intensitätsmaß</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Als <b>Intensitätsmaß</b> bezeichnet man in der Mathematik ein <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Maß</a>, das einem <a href="Zuf%C3%A4lliges_Ma%C3%9F" title="Zufälliges Maß">zufälligen Maß</a> zugeordnet wird. Das Intensitätsmaß entspricht dabei dem <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> des zufälligen Maßes und gibt somit an, welches Volumen das zufällige Maß im Schnitt einer gewissen Menge zuordnet. Das Intensitätsmaß enthält somit wichtige Informationen über das zufällige Maß. So sind beispielsweise <a href="Poisson-Prozess" title="Poisson-Prozess">Poisson-Prozesse</a> durch die Angabe ihres Intensitätsmaßes bereits eindeutig bestimmt.

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

<p>Gegeben sei ein <a href="Zuf%C3%A4lliges_Ma%C3%9F" title="Zufälliges Maß">zufälliges Maß</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.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> auf dem <a href="Messraum_(Mathematik)" title="Messraum (Mathematik)">Messraum</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 (S,{\mathcal {A}})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b3e86459e162dd204c4be9b6fc28fab52a208ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.246ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {A}})}" loading="lazy"></span>. Das bedeutet, dass <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.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> fast sicher lokal endliche Maße auf <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 (S,{\mathcal {A}})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b3e86459e162dd204c4be9b6fc28fab52a208ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.246ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {A}})}" loading="lazy"></span> als Werte annimmt.

</p><p>Dann heißt das Maß <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8554da649f73451c799abbca923279c631b765c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.95ex; height:2.176ex;" alt="{\displaystyle \operatorname {E} X}" loading="lazy"></span> auf <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 (S,{\mathcal {A}})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b3e86459e162dd204c4be9b6fc28fab52a208ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.246ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {A}})}" loading="lazy"></span>, das durch

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} X\colon \left\{{\begin{aligned}{\mathcal {A}}&\to [0,\infty ]\\A&\mapsto \operatorname {E} [X(A)]\end{aligned}}\right.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eeb8a20d35604559e978dd7fac75ac48182cfca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.405ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} X\colon \left\{{\begin{aligned}{\mathcal {A}}&\to [0,\infty ]\\A&\mapsto \operatorname {E} [X(A)]\end{aligned}}\right.}" loading="lazy"></span></dd></dl>

<p>gegeben ist, das Intensitätsmaß von <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.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>.<sup id="cite_ref-Klenke552_1-0" class="reference"><a href="#cite_note-Klenke552-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Hierbei ist zu unterscheiden zwischen der Bezeichnung des Intensitätsmaßes als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8554da649f73451c799abbca923279c631b765c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.95ex; height:2.176ex;" alt="{\displaystyle \operatorname {E} X}" loading="lazy"></span> und der Bildung des Erwartungswertes einer Zufallsvariable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [Y]}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/639e8577c6faffc0471c7e123ead30970034e6d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [Y]}" loading="lazy"></span>.

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

<p>Bei einem <a href="Binomial-Prozess" title="Binomial-Prozess">Binomial-Prozess</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.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> gegeben durch <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 n}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> und eine Verteilung <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 \nu }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span> gilt per Konstruktion <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(A)\sim \operatorname {Bin} _{n,\nu (A)}}">

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<annotation encoding="application/x-tex">{\displaystyle X(A)\sim \operatorname {Bin} _{n,\nu (A)}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d72a048cdcbdc0836f88d09cb9dfc0577241856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.275ex; height:3.176ex;" alt="{\displaystyle X(A)\sim \operatorname {Bin} _{n,\nu (A)}}" loading="lazy"></span>. Mit den elementaren Eigenschaften der Binomialverteilung folgt dann direkt

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} X(A)=\operatorname {E} [X(A)]=n\nu (A)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43ec4d0d073a016fed7359ff7d9e45716488339c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.287ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} X(A)=\operatorname {E} [X(A)]=n\nu (A)}" loading="lazy"></span>.</dd></dl>

<p>Also ist das Intensitätsmaß eines Binomialprozesses gegeben durch

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fe4d85a4a5eff170feae28ba4f18d82b589d6a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.675ex; height:2.176ex;" alt="{\displaystyle \operatorname {E} X=n\nu }" loading="lazy"></span>.</dd></dl>

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

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

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8554da649f73451c799abbca923279c631b765c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.95ex; height:2.176ex;" alt="{\displaystyle \operatorname {E} X}" loading="lazy"></span> ist stets <a href="S-finites_Ma%C3%9F" title="S-finites Maß">s-finit</a> und erfüllt

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[\int f(x)X(\mathrm {d} x)\right]=\int f(x)\operatorname {E} X(dx)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e9f6801ec4bf06ca3aa41022fd8895521ba8a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.236ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} \left[\int f(x)X(\mathrm {d} x)\right]=\int f(x)\operatorname {E} X(dx)}" loading="lazy"></span></dd></dl>

<p>für jede positive messbare Funktion auf <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 (S,{\mathcal {A}})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b3e86459e162dd204c4be9b6fc28fab52a208ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.246ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {A}})}" loading="lazy"></span>.<sup id="cite_ref-Kallenberg53_2-0" class="reference"><a href="#cite_note-Kallenberg53-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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

<ul><li>Olav Kallenberg: <cite style="font-style:italic">Random Measures, Theory and Applications</cite>. Springer, Switzerland 2017, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-41598-7">10.1007/978-3-319-41598-7</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Intensit%C3%A4tsma%C3%9F&rft.au=Olav+Kallenberg&rft.btitle=Random+Measures%2C+Theory+and+Applications&rft.date=2017&rft.doi=10.1007%2F978-3-319-41598-7&rft.genre=book&rft.place=Switzerland&rft.pub=Springer" style="display:none"> </span></li>

<li>Achim Klenke: <cite style="font-style:italic">Wahrscheinlichkeitstheorie</cite>. 3. Auflage. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-36018-3">10.1007/978-3-642-36018-3</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Intensit%C3%A4tsma%C3%9F&rft.au=Achim+Klenke&rft.btitle=Wahrscheinlichkeitstheorie&rft.date=2013&rft.doi=10.1007%2F978-3-642-36018-3&rft.edition=3.&rft.genre=book&rft.isbn=9783642360176&rft.place=Berlin+Heidelberg&rft.pub=Springer-Verlag" style="display:none"> </span></li></ul>

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

<li id="cite_note-Klenke552-1"><span class="mw-cite-backlink"><a href="#cite_ref-Klenke552_1-0">↑</a></span> <span class="reference-text"> Achim Klenke: <cite style="font-style:italic">Wahrscheinlichkeitstheorie</cite>. 3. Auflage. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-36018-3">10.1007/978-3-642-36018-3</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Intensit%C3%A4tsma%C3%9F&rft.au=Achim+Klenke&rft.btitle=Wahrscheinlichkeitstheorie&rft.date=2013&rft.doi=10.1007%2F978-3-642-36018-3&rft.edition=3.&rft.genre=book&rft.isbn=9783642360176&rft.place=Berlin+Heidelberg&rft.pub=Springer-Verlag" style="display:none"> </span> </span>

</li>

<li id="cite_note-Kallenberg53-2"><span class="mw-cite-backlink"><a href="#cite_ref-Kallenberg53_2-0">↑</a></span> <span class="reference-text"> Olav Kallenberg: <cite style="font-style:italic">Random Measures, Theory and Applications</cite>. Springer, Switzerland 2017, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>53</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-41598-7">10.1007/978-3-319-41598-7</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Intensit%C3%A4tsma%C3%9F&rft.au=Olav+Kallenberg&rft.btitle=Random+Measures%2C+Theory+and+Applications&rft.date=2017&rft.doi=10.1007%2F978-3-319-41598-7&rft.genre=book&rft.pages=53&rft.place=Switzerland&rft.pub=Springer" style="display:none"> </span></span>

</li>

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