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<title>Diracmaß</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Diracmaß</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">Ein Diracmaß, benannt nach dem Physiker <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, ist ein <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Maß</a> in der <a href="Ma%C3%9Ftheorie" title="Maßtheorie">Maßtheorie</a> mit einelementigem <a href="Tr%C3%A4ger_(Ma%C3%9Ftheorie)" title="Träger (Maßtheorie)">Träger</a>. Das Diracmaß ist die Verteilung einer <a href="Fast_sicher" title="Fast sicher">fast sicher</a> konstanten <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariable</a> und spielt eine Rolle als Formalisierung des Begriffes der <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a>.

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

Es sei ein <a href="Messbarer_Raum" class="mw-redirect" title="Messbarer Raum">messbarer Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/529e4b59b55bf6c1b052a0a3c51f90aaa175b912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.425ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}})}" loading="lazy"> gegeben, also eine Grundmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> zusammen mit einer darauf definierten <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy">. Zu jedem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34361065905ff8a9a986d2c0dc8010e769c8e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \Omega }" loading="lazy"> wird eine zugehörige Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fee6a037fd2068de909aed167e1f41ea8775169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.676ex;" alt="{\displaystyle \delta _{z}}" loading="lazy"> definiert, die jeder Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {A}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6e69647797be244cf2ebc28ecd61fafba8790c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {A}}}" loading="lazy"> den Wert <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.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"> zuordnet, wenn sie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> enthält, und den Wert <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.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">, wenn sie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> nicht enthält:

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<mtext>falls </mtext>

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<mtext>sonst.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}(A):={\begin{cases}1,&{\text{falls }}z\in A,\\0&{\text{sonst.}}\end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64e41cc163f60c82b6d26caa7177c8a21402e8ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.942ex; height:6.176ex;" alt="{\displaystyle \delta _{z}(A):={\begin{cases}1,&{\text{falls }}z\in A,\\0&{\text{sonst.}}\end{cases}}}" loading="lazy"></dd></dl>

Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}\colon {\mathcal {A}}\to [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}\colon {\mathcal {A}}\to [0,1]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74d5f682a2d1d77e211432e697cafc862f9f0f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.238ex; height:2.843ex;" alt="{\displaystyle \delta _{z}\colon {\mathcal {A}}\to [0,1]}" loading="lazy"> ist dann ein <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Maß</a> und wird Diracmaß oder Punktmaß im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> genannt. Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}(\Omega )=1}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}(\Omega )=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e21ecebb1efae18b87d6592812ff5f604b96d872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.782ex; height:2.843ex;" alt="{\displaystyle \delta _{z}(\Omega )=1}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fee6a037fd2068de909aed167e1f41ea8775169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.676ex;" alt="{\displaystyle \delta _{z}}" loading="lazy"> sogar ein <a href="Wahrscheinlichkeitsma%C3%9F" title="Wahrscheinlichkeitsmaß">Wahrscheinlichkeitsmaß</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}},\delta _{z})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5beaf8e90a9e6f60d4d688b327cf97a38213415c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.493ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},\delta _{z})}" loading="lazy"> ein <a href="Wahrscheinlichkeitsraum" title="Wahrscheinlichkeitsraum">Wahrscheinlichkeitsraum</a>. Damit lässt sich die <a href="Dirac-Verteilung" title="Dirac-Verteilung">Dirac-Verteilung</a> definieren. Beim Diracmaß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fee6a037fd2068de909aed167e1f41ea8775169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.676ex;" alt="{\displaystyle \delta _{z}}" loading="lazy"> ist die Einheitsmasse im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> konzentriert. Es folgt, dass das Maß endlich ist, insbesondere ist der Maßraum <a href="%CE%A3-endlich" class="mw-redirect" title="Σ-endlich">σ-endlich</a>.

Mit Hilfe der <a href="Indikatorfunktion" title="Indikatorfunktion">charakteristischen Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> kann man die definierende Gleichung auch durch

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}(A)=\chi _{A}(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d656650e0850c64945e92e13392ce95c92db235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.502ex; height:2.843ex;" alt="{\displaystyle \delta _{z}(A)=\chi _{A}(z)}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34361065905ff8a9a986d2c0dc8010e769c8e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \Omega }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {A}}}">

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</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6e69647797be244cf2ebc28ecd61fafba8790c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {A}}}" loading="lazy"> ausdrücken.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_des_Dirac-Maßes">Eigenschaften des Dirac-Maßes</h2></div>

<msub>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> sei das Dirac-Maß, das auf einem festen Punkt <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in einem messbaren Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\Sigma )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63c4fcf15808c3dbf52cc4b3fd473e97e231f53f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (X,\Sigma )}" loading="lazy"> zentriert ist.

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> ist ein <a href="Wahrscheinlichkeitsma%C3%9F" title="Wahrscheinlichkeitsmaß">Wahrscheinlichkeitsmaß</a> und damit ein endliches Maß.</li></ul>

Angenommen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,T)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/627ce34fb2db641fafd1e88cf307a668d1789ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle (X,T)}" loading="lazy"> ein <a href="Topologischer_Raum" title="Topologischer Raum">topologischer Raum</a> ist und dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> mindestens so fein wie die borelsche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (T)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9888a5496892a8c482ad465f48b839102187bd1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle \sigma (T)}" loading="lazy"> auf <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><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"> ist, dann gilt:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> ist ein streng positives Maß dann und nur dann, wenn die <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> so ist, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> innerhalb jeder nichtleeren offenen Menge liegt, z. B. im Fall der trivialen Topologie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varnothing ,X\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a150d87fcc80740568e40c058f8c33c84b626465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.147ex; height:2.843ex;" alt="{\displaystyle \{\varnothing ,X\}}" loading="lazy">.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> ein Wahrscheinlichkeitsmaß ist, ist es auch ein lokal endliches Maß.</li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein topologischer <a href="Hausdorff-Raum" title="Hausdorff-Raum">Hausdorff-Raum</a> mit seiner borelschen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-Algebra ist, dann erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> die Eigenschaft, ein inneres <a href="Regul%C3%A4res_Ma%C3%9F" title="Reguläres Maß">reguläres Maß</a> zu sein, da Einermengen wie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"> immer kompakt sind. Folglich ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> auch ein <a href="Radon-Ma%C3%9F" class="mw-redirect" title="Radon-Maß">Radon-Maß</a>.</li>

<li>Unter der Annahme, dass die Topologie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> fein genug ist, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"> abgeschlossen ist (was in den meisten Anwendungen der Fall ist), ist der <a href="Tr%C3%A4ger_(Ma%C3%9Ftheorie)" title="Träger (Maßtheorie)">Träger</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy">. (Andernfalls ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {supp} \left(\delta _{x}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {supp} \left(\delta _{x}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9309c562f2c1427e3299b58f3ce15cbc7c058dac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.808ex; height:2.843ex;" alt="{\displaystyle \operatorname {supp} \left(\delta _{x}\right)}" loading="lazy"> der Abschluss von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,T)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/627ce34fb2db641fafd1e88cf307a668d1789ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle (X,T)}" loading="lazy">.) Außerdem ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> das einzige Wahrscheinlichkeitsmaß, dessen Träger <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a120eeb8a091b516595765bd08b306f2394e7721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}" loading="lazy"> ist.</li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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">-dimensionale <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidische Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> mit seiner üblichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-Algebra und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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">-dimensionalem <a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6ad7478302bf9f9f9d5900ebb420994813946ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.574ex; height:2.343ex;" alt="{\displaystyle \lambda ^{n}}" loading="lazy"> ist, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"> ein <a href="Singul%C3%A4res_Ma%C3%9F" title="Singuläres Maß">singuläres Maß</a> in Bezug auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6ad7478302bf9f9f9d5900ebb420994813946ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.574ex; height:2.343ex;" alt="{\displaystyle \lambda ^{n}}" loading="lazy">: Man zerlege einfach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\mathbb {R} ^{n}\setminus \{x\}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e062f5695774e093e7829df7d2e6ee896865ee80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.587ex; height:2.843ex;" alt="{\displaystyle A=\mathbb {R} ^{n}\setminus \{x\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{x\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16a8bd2b763f0b6509ea8303801d61105aecaf0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.517ex; height:2.843ex;" alt="{\displaystyle B=\{x\}}" loading="lazy"> und stelle fest, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{x}(A)=\lambda ^{n}(B)=0}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{x}(A)=\lambda ^{n}(B)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e10dfca73d301cf63cdec70bc8918f5bd961ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.264ex; height:2.843ex;" alt="{\displaystyle \delta _{x}(A)=\lambda ^{n}(B)=0}" loading="lazy">.</li>

<li>Das Dirac-Maß ist ein <a href="%CE%A3-Endlichkeit" title="Σ-Endlichkeit"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-endliches Maß.</a></li></ul>

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

<div class="sieheauch" role="navigation" style="font-style:italic;">Siehe auch: <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a></div>

Das Dirac-Integral der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e825429becf7730acee7e5a9da089c6b98ca759d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.348ex; height:2.509ex;" alt="{\displaystyle f\colon A\to \mathbb {R} }" loading="lazy"> ist definiert als das <a href="Lebesgue-Integral" title="Lebesgue-Integral">Lebesgue-Integral</a> bezüglich des Dirac-Maßes. Anstelle des Lebesgue-Maßes wird zur Berechnung des Integrals das Dirac-Maß verwendet. Damit ergibt sich für das Integral einer beliebigen Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">:

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{A}f\mathrm {d} \delta _{z}={\begin{cases}f(z)&z\in A\\0&z\notin A\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132d5b83a43682185118969568beae9f82f0652e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.513ex; height:6.176ex;" alt="{\displaystyle \int _{A}f\mathrm {d} \delta _{z}={\begin{cases}f(z)&z\in A\\0&z\notin A\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Begründung">Begründung</h3></div>

Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e825429becf7730acee7e5a9da089c6b98ca759d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.348ex; height:2.509ex;" alt="{\displaystyle f\colon A\to \mathbb {R} }" loading="lazy"> sei eine nicht-negative <a href="Messbare_Funktion" title="Messbare Funktion">messbare Funktion</a>. Das Lebesgue-Integral der Funktion bezüglich des Dirac-Maßes ist durch

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{A}f\mathrm {d} \delta _{z}=\lim _{n\rightarrow \infty }\int _{A}f_{n}\mathrm {d} \delta _{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d67a3f4f2096c6faa36b4de435a405fb7fb6846.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.336ex; height:5.676ex;" alt="{\displaystyle \int _{A}f\mathrm {d} \delta _{z}=\lim _{n\rightarrow \infty }\int _{A}f_{n}\mathrm {d} \delta _{z}}" loading="lazy"></dd></dl>

definiert, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> eine beliebige Folge nichtnegativer <a href="Einfache_Funktion" title="Einfache Funktion">einfacher Funktionen</a> ist, die punktweise und monoton wachsend gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> konvergiert. Eine einfache Funktion ist eine messbare Funktion, die nur endlich viele Funktionswerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b1fb627423abe4988b7ed88d4920bf1ec074790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}" loading="lazy"> annimmt. <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"> sei die Anzahl der Funktionswerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b1fb627423abe4988b7ed88d4920bf1ec074790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}" loading="lazy">; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

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</mrow>

</msub>

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> seien die (messbaren) Mengen, auf der die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

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</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> jeweils den Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b1fb627423abe4988b7ed88d4920bf1ec074790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}" loading="lazy"> annimmt. Das Integral einer einfachen Funktion ist damit folgendermaßen definiert:

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle \int _{A}f_{n}\mathrm {d} \delta _{z}=\sum _{i=1}^{m}\alpha _{i}(n)\delta _{z}(A_{i}(n))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b722ab87a1d5b4c8f6b422c4597c876bb740057b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.751ex; height:6.843ex;" alt="{\displaystyle \int _{A}f_{n}\mathrm {d} \delta _{z}=\sum _{i=1}^{m}\alpha _{i}(n)\delta _{z}(A_{i}(n))}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\notin A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75d125014903489672d365cefe0bca8b71338eb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.672ex; height:2.676ex;" alt="{\displaystyle z\notin A}" loading="lazy">, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> erst recht nicht Element irgendeiner der Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy">. Dann ist auch das Dirac-Maß von allen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> gleich null. Folglich ist das Integral über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" 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 A}" loading="lazy"> insgesamt gleich null.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in A_{j}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle z\in A_{j}(n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/036abcb463ba92bfae813159ed69f15c27feb407.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.786ex; height:3.009ex;" alt="{\displaystyle z\in A_{j}(n)}" loading="lazy"> für irgendein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">, so ist das Dirac-Maß von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}(n)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/027f6c49596b3ffe4742fa28ca5fd76bae89c303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.857ex; height:3.009ex;" alt="{\displaystyle A_{j}(n)}" loading="lazy"> gleich <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.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">; das Dirac-Maß für alle anderen Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}(n)}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f54accfcb5e1ad0e0f3de904b9a3de811336dc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.747ex; height:2.843ex;" alt="{\displaystyle A_{i}(n)}" loading="lazy"> ist dann gleich null. Für das Integral der einfachen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> ergibt sich somit:

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<annotation encoding="application/x-tex">{\displaystyle \int _{A}f_{n}\mathrm {d} \delta _{z}=\alpha _{j}(n)=f_{n}(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c332810c1c416ec520f246d24fa0aab0cf2dc0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.882ex; height:5.676ex;" alt="{\displaystyle \int _{A}f_{n}\mathrm {d} \delta _{z}=\alpha _{j}(n)=f_{n}(z)}" loading="lazy"></dd>

</mrow>

</munder>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }\int _{A}f_{n}\mathrm {d} \delta _{z}=\lim _{n\rightarrow \infty }f_{n}(z)=f(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68b73b172bea1d590cbcdf656b4e667567b6f471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.776ex; height:5.676ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }\int _{A}f_{n}\mathrm {d} \delta _{z}=\lim _{n\rightarrow \infty }f_{n}(z)=f(z)}" loading="lazy"></dd></dl>

Also ist das Dirac-Integral gleich dem Funktionswert an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40028fe140831e54037c7b9f8d6907fdeaf4c256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle z\in A}" loading="lazy"> ist.

Eine andere Beweisführung erfolgt so:

Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34361065905ff8a9a986d2c0dc8010e769c8e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \Omega }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {A}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6e69647797be244cf2ebc28ecd61fafba8790c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {A}}}" loading="lazy"> gilt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{A}f\,\mathrm {d} \delta _{z}&=\int _{A\cap f^{-1}(\{f(z)\})}f\,\mathrm {d} \delta _{z}+\int _{A\setminus f^{-1}(\{f(z)\})}f\,\mathrm {d} \delta _{z}\\&=\int _{\{x\in A\vert f(x)=f(z)\}}f\,\mathrm {d} \delta _{z}+\int _{\{x\in A\mid f(x)\neq f(z)\}}f\,\mathrm {d} \delta _{z}\\&=f(z)\delta _{z}(A)+0\\&=f(z)\delta _{z}(A)\\&={\begin{cases}f(z)&z\in A\\0&z\not \in A\end{cases}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd853aec37673705317b3ebaabee4390a9a911e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.704ex; margin-bottom: -0.3ex; width:54.086ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}\int _{A}f\,\mathrm {d} \delta _{z}&=\int _{A\cap f^{-1}(\{f(z)\})}f\,\mathrm {d} \delta _{z}+\int _{A\setminus f^{-1}(\{f(z)\})}f\,\mathrm {d} \delta _{z}\\&=\int _{\{x\in A\vert f(x)=f(z)\}}f\,\mathrm {d} \delta _{z}+\int _{\{x\in A\mid f(x)\neq f(z)\}}f\,\mathrm {d} \delta _{z}\\&=f(z)\delta _{z}(A)+0\\&=f(z)\delta _{z}(A)\\&={\begin{cases}f(z)&z\in A\\0&z\not \in A\end{cases}}\end{aligned}}}" loading="lazy"></dd></dl>

Als einelementige Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f(z)\}\in {\mathcal {B}}}">

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</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d4253f15bd1d4c642528666856b5d005bae1608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.885ex; height:2.843ex;" alt="{\displaystyle \{f(z)\}\in {\mathcal {B}}}" loading="lazy">. Urbilder messbarer Mengen sind messbar. Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\{f(z)\})\in {\mathcal {A}}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\{f(z)\})\in {\mathcal {A}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74dff1ca4ae6f25b5bab7813ed7dc9ffabe6d59c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.708ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\{f(z)\})\in {\mathcal {A}}}" loading="lazy"> und dementsprechend auch die Mengen, über die oben integriert wird.

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z\}\in {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb3bf2bb598c341e92e54b2dfe18c6ec03a8e35a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.157ex; height:2.843ex;" alt="{\displaystyle \{z\}\in {\mathcal {A}}}" loading="lazy">, so ist auch eine Integration über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap \{z\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b07f3290e20cfe5ec9476c24fbf1855ab38df627.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.739ex; height:2.843ex;" alt="{\displaystyle A\cap \{z\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus \{z\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33a160e9614ad4894df37cd29c0fed258e075a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.351ex; height:2.843ex;" alt="{\displaystyle A\setminus \{z\}}" loading="lazy"> möglich.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Z%C3%A4hlma%C3%9F_(Ma%C3%9Ftheorie)" title="Zählmaß (Maßtheorie)">Zählmaß</a></li>

<li><a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a></li>

<li><a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a></li></ul>

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

<ul><li><a href="Elliott_H._Lieb" class="mw-redirect" title="Elliott H. Lieb">Elliott H. Lieb</a>, <a href="Michael_Loss" title="Michael Loss">Michael Loss</a>: Analysis (= Graduate Studies in Mathematics. Bd. 14). 2nd Edition. American Mathematical Society, Providence RI 2001, ISBN 0-8218-2783-9.</li>

<li>Jean Dieudonnè 1976: Treatise on analysis, Part 2 (Seite 100), ISBN 0-12-215502-5</li>

<li>Benedetto, John (1997): <a rel="nofollow" class="external text" href="https://books.google.at/books?id=_SCeYgvPgoYC&pg=PA72&redir_esc=y#v=onepage&q&f=false">"§2.1.3 Definition, δ"</a> Harmonic analysis and applications. CRC Press (Seite 72), ISBN 0-8493-7879-6</li></ul></div><div><div class="zim-footer">

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