English+GermanWikipedia - /page/entry.mu

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<title>Bochner-Integral</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Bochner-Integral</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">Das Bochner-Integral, benannt nach <a href="Salomon_Bochner" title="Salomon Bochner">Salomon Bochner</a>, ist eine Verallgemeinerung des <a href="Lebesgue-Integral" title="Lebesgue-Integral">Lebesgue-Integrals</a> auf <a href="Banachraum" title="Banachraum">Banachraum</a>-wertige Funktionen. Das Integral lässt sich aber auch auf <a href="Lokalkonvexer_Raum" title="Lokalkonvexer Raum">lokalkonvexe Räume</a> verallgemeinern.

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

Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}},\mu )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cabc7dd1ffb04becc182e5356682ec027db7211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},\mu )}" loading="lazy"> ein <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">-endlicher, <a href="Vollst%C3%A4ndiger_Ma%C3%9Fraum" class="mw-redirect" title="Vollständiger Maßraum">vollständiger Maßraum</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B,\|\cdot \|)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d8873f311a155a76f4abcef9649478b53911372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.611ex; height:2.843ex;" alt="{\displaystyle (B,\|\cdot \|)}" loading="lazy"> ein <a href="Banachraum" title="Banachraum">Banachraum</a>.

Das Bochner-Integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }f\,{\rm {d}}\mu }">

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<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f\,{\rm {d}}\mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/146a7a12cf6d59aa72ec5b20089ee743b4ec8acd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.458ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f\,{\rm {d}}\mu }" loading="lazy"> einer Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \to B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7371a27340518b4ba3fb0355ddb8d8834014c6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.369ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \to B}" loading="lazy"> ist nun folgendermaßen definiert:

Als <a href="Einfache_Funktion" title="Einfache Funktion">einfache Funktion</a> bezeichnen wir Funktionen der Gestalt

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<annotation encoding="application/x-tex">{\displaystyle s(x)=\sum _{i=1}^{m}\alpha _{i}\chi _{X_{i}}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9636b99f81bb4c48122090cbf790112ce72109a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.169ex; height:6.843ex;" alt="{\displaystyle s(x)=\sum _{i=1}^{m}\alpha _{i}\chi _{X_{i}}(x)}" loading="lazy"></dd></dl>

mit Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}\in B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d346a5217f4674b63f5e20fee84be25f35f8de5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.892ex; height:2.509ex;" alt="{\displaystyle \alpha _{i}\in B}" loading="lazy"> und messbaren Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}\in {\mathcal {A}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cd529a4795491e94181a9903c9c14f9e1287a03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.676ex;" alt="{\displaystyle X_{i}\in {\mathcal {A}}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{X_{i}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/347f5ae669ab3944aaf6367b74a7b3e0e9a4e6f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle \chi _{X_{i}}}" loading="lazy"> deren <a href="Indikatorfunktion" title="Indikatorfunktion">Indikatorfunktion</a> bezeichnet. Das Integral einer einfachen Funktion ist nun auf naheliegende Weise definiert:

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<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }s\,{\rm {d}}\mu :=\sum _{i=1}^{m}\alpha _{i}\mu (X_{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05afbb4025b62d2c591957f1167dea3296482b51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.98ex; height:6.843ex;" alt="{\displaystyle \int _{\Omega }s\,{\rm {d}}\mu :=\sum _{i=1}^{m}\alpha _{i}\mu (X_{i})}" loading="lazy">,</dd></dl>

wobei dies wohldefiniert, also unabhängig von der konkreten Zerlegung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> ist.<a href="#cite_note-1">[1]</a>

Eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \rightarrow B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c73afdbfc61a058890d655081042b4facebfe2f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.369ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \rightarrow B}" loading="lazy"> heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-messbar oder Bochner-messbar, wenn es eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s_{n})_{n\in \mathbb {N} }}">

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<annotation encoding="application/x-tex">{\displaystyle (s_{n})_{n\in \mathbb {N} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294c2aff6dc4227716807e2d26f0aa31f0115db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.62ex; height:2.843ex;" alt="{\displaystyle (s_{n})_{n\in \mathbb {N} }}" loading="lazy"> einfacher Funktionen gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lim _{n\to \infty }s_{n}(x)=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \lim _{n\to \infty }s_{n}(x)=f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4123f8e1cbf863961ec94866f0f95337ca9f9677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \lim _{n\to \infty }s_{n}(x)=f(x)}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40de8fa70c126001826d34da2a114349d8476bb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \Omega }" loading="lazy"> gilt.<a href="#cite_note-2">[2]</a>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-messbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \rightarrow B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c73afdbfc61a058890d655081042b4facebfe2f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.369ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \rightarrow B}" loading="lazy"> heißt Bochner-integrierbar<a href="#cite_note-3">[3]</a>, falls es eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s_{n})_{n\in \mathbb {N} }}">

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<annotation encoding="application/x-tex">{\displaystyle (s_{n})_{n\in \mathbb {N} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294c2aff6dc4227716807e2d26f0aa31f0115db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.62ex; height:2.843ex;" alt="{\displaystyle (s_{n})_{n\in \mathbb {N} }}" loading="lazy"> einfacher Funktionen gibt, so dass

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }s_{n}(x)=f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2261d425e08c0a4bf7a47060baf4e89fbf0e096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.624ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }s_{n}(x)=f(x)}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40de8fa70c126001826d34da2a114349d8476bb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \Omega }" loading="lazy"> gilt und</li>

<li>zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}=n_{0}(\varepsilon )\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n_{0}=n_{0}(\varepsilon )\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/728483929739c710b1da782c217acc24bd1a3afe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.408ex; height:2.843ex;" alt="{\displaystyle n_{0}=n_{0}(\varepsilon )\in \mathbb {N} }" loading="lazy"> existiert mit</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\|s_{n}-s_{k}\|{\rm {d}}\mu <\varepsilon }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ac4c28df60183b8e57768b72b20b2563e623648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.628ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\|s_{n}-s_{k}\|{\rm {d}}\mu <\varepsilon }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,k\geq n_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle n,k\geq n_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2336818a4da6e22d8e93d471bea6570cb97c3e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.187ex; height:2.509ex;" alt="{\displaystyle n,k\geq n_{0}}" loading="lazy">.</dd></dl></dd></dl>

In diesem Fall ist

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<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }f\,{\rm {d}}\mu :=\lim _{n\to \infty }\int _{\Omega }s_{n}\,{\rm {d}}\mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccece7d4917d410921c00ce1191d3a168e70b436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.352ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }f\,{\rm {d}}\mu :=\lim _{n\to \infty }\int _{\Omega }s_{n}\,{\rm {d}}\mu }" loading="lazy"></dd></dl>

wohldefiniert, das heißt unabhängig von der Wahl der konkreten Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s_{n})_{n\in \mathbb {N} }}">

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<annotation encoding="application/x-tex">{\displaystyle (s_{n})_{n\in \mathbb {N} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294c2aff6dc4227716807e2d26f0aa31f0115db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.62ex; height:2.843ex;" alt="{\displaystyle (s_{n})_{n\in \mathbb {N} }}" loading="lazy"> mit obigen Eigenschaften.<a href="#cite_note-4">[4]</a>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a06daa7718684350114b25680319f138f618d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.186ex; height:2.343ex;" alt="{\displaystyle M\in {\mathcal {A}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4ad89280646640f9640616bb426ddb57f3e4f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.133ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow B}" loading="lazy">, so schreibt man

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<annotation encoding="application/x-tex">{\displaystyle \int _{M}f{\rm {d}}\mu :=\int _{\Omega }{\tilde {f}}{\rm {d}}\mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/782cd2beff3356e48fc63ca940ba943066903e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.849ex; height:5.676ex;" alt="{\displaystyle \int _{M}f{\rm {d}}\mu :=\int _{\Omega }{\tilde {f}}{\rm {d}}\mu }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}(x):=\left\{{\begin{array}{ll}f(x)\ ,&{\rm {falls}}\ x\in M\ ,\\0\ ,&{\rm {falls}}\ x\in \Omega \setminus M,\\\end{array}}\right.}">

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(x):=\left\{{\begin{array}{ll}f(x)\ ,&{\rm {falls}}\ x\in M\ ,\\0\ ,&{\rm {falls}}\ x\in \Omega \setminus M,\\\end{array}}\right.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1666e1b309283d5105cf9bd56e29ed6da045bc04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.996ex; height:6.176ex;" alt="{\displaystyle {\tilde {f}}(x):=\left\{{\begin{array}{ll}f(x)\ ,&{\rm {falls}}\ x\in M\ ,\\0\ ,&{\rm {falls}}\ x\in \Omega \setminus M,\\\end{array}}\right.}" loading="lazy"></dd></dl>

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

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6cb99679a4b79cb5ca3c242811bd91220c91f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}}" loading="lazy"> Bochner-integrierbar ist.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Messbarkeitssatz_von_Pettis">Messbarkeitssatz von Pettis</h2></div>

Der folgende auf <a href="Billy_James_Pettis" title="Billy James Pettis">Billy James Pettis</a> zurückgehende Satz charakterisiert die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-Messbarkeit:

Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7371a27340518b4ba3fb0355ddb8d8834014c6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.369ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \to B}" loading="lazy"> ist genau dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-messbar, wenn die folgenden beiden Bedingungen erfüllt sind:

<ul><li>Für jedes stetige <a href="Lineares_Funktional" class="mw-redirect" title="Lineares Funktional">lineare Funktional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in B'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de6d93092f1adcf6ce3b7c810e5beec50fb61263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.675ex; height:2.843ex;" alt="{\displaystyle \phi \in B'}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \circ f\colon \Omega \to {\mathbb {K} }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \circ f\colon \Omega \to {\mathbb {K} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1579bbbb5de56148959bdbaf0a55caf1e98750f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.993ex; height:2.509ex;" alt="{\displaystyle \phi \circ f\colon \Omega \to {\mathbb {K} }}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-messbar.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-<a href="Nullmenge" title="Nullmenge">Nullmenge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\subset \Omega }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65b9f9b11e2fa65f2db63485ca822bba118fd29c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.84ex; height:2.176ex;" alt="{\displaystyle N\subset \Omega }" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Omega \setminus N)\subset B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\Omega \setminus N)\subset B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c03b477654b0b2b39d151c817344743944f7eb11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.887ex; height:2.843ex;" alt="{\displaystyle f(\Omega \setminus N)\subset B}" loading="lazy"> <a href="Separabler_Raum" title="Separabler Raum">separabel</a> bzgl. der <a href="Normtopologie" title="Normtopologie">Normtopologie</a> ist.</li></ul>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ein separabler Banachraum, so ist die zweite Bedingung automatisch erfüllt und damit entbehrlich. Insgesamt ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-Messbarkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">-wertiger Funktionen mit diesem Satz auf die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-Messbarkeit skalarer Funktionen zurückgeführt.

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

Die folgende von Bochner gefundene äquivalente Charakterisierung Bochner-integrierbarer Funktionen erlaubt es, einige klassische Resultate der lebesgueschen Integrationstheorie wie z. B. den <a href="Satz_von_der_majorisierten_Konvergenz" title="Satz von der majorisierten Konvergenz">Satz von der majorisierten Konvergenz</a> auf das Bochner-Integral zu übertragen:

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-messbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \to B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7371a27340518b4ba3fb0355ddb8d8834014c6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.369ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \to B}" loading="lazy"> ist genau dann Bochner-integrierbar, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|:\Omega \to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|:\Omega \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b7d46cb50ea28fd4eaf5031dcbb86d3ec1c0ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.511ex; height:2.843ex;" alt="{\displaystyle \|f\|:\Omega \to \mathbb {R} }" loading="lazy"> Lebesgue-integrierbar ist.

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

In diesem Abschnitt ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ein Banachraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon \Omega \rightarrow B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g\colon \Omega \rightarrow B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e17076ae177834aade54c681de0bfc162c44cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.519ex; height:2.509ex;" alt="{\displaystyle f,g\colon \Omega \rightarrow B}" loading="lazy"> sind integrierbare Funktionen.

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

Das Bochner-Integral ist <a href="Lineare_Abbildung" title="Lineare Abbildung">linear</a>, das heißt, für Bochner-integrierbare Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon \Omega \rightarrow B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g\colon \Omega \rightarrow B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e17076ae177834aade54c681de0bfc162c44cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.519ex; height:2.509ex;" alt="{\displaystyle f,g\colon \Omega \rightarrow B}" loading="lazy"> und beliebige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in \mathbb {K} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in \mathbb {K} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f23202f05556aa217e4bafea0e158499880f0c03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.502ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta \in \mathbb {K} }" loading="lazy"> ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha f+\beta g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76d96425f225ba736f51a9e03491467de6f4109c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.055ex; height:2.509ex;" alt="{\displaystyle \alpha f+\beta g}" loading="lazy"> integrierbar, und es gilt:

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }(\alpha f+\beta g)\,\mathrm {d} \mu =\alpha \int _{\Omega }f\,\mathrm {d} \mu +\beta \int _{\Omega }g\,\mathrm {d} \mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53d9113cccd1be2afd6577ae027256b877e1e854.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.943ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }(\alpha f+\beta g)\,\mathrm {d} \mu =\alpha \int _{\Omega }f\,\mathrm {d} \mu +\beta \int _{\Omega }g\,\mathrm {d} \mu }" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Verkettung_mit_einem_stetigen_Operator">Verkettung mit einem stetigen Operator</h3></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> ein Banachraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in L(B,D)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\in L(B,D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5bce4ee2a07c668735e1af7ccbc2b8a990a51b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.591ex; height:2.843ex;" alt="{\displaystyle T\in L(B,D)}" loading="lazy"> ein <a href="Linearer_Operator" title="Linearer Operator">stetiger linearer Operator</a>. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tf\colon \Omega \to D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e42144f0870ff01a38d70d9f9708603bb8c14ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.165ex; height:2.509ex;" alt="{\displaystyle Tf\colon \Omega \to D}" loading="lazy"> eine integrierbare Funktion und es gilt<a href="#cite_note-6">[6]</a>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\left(\int _{\Omega }f(x)\mathrm {d} \mu (x)\right)=\int _{\Omega }T(f(x))\mathrm {d} \mu (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2bc9b8d96d2275a741cd50a247061a4c80e549c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.687ex; height:6.176ex;" alt="{\displaystyle T\left(\int _{\Omega }f(x)\mathrm {d} \mu (x)\right)=\int _{\Omega }T(f(x))\mathrm {d} \mu (x)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Radon–Nikodym-Eigenschaft">Radon–Nikodym-Eigenschaft</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Radon-Nikodym-Eigenschaft" title="Radon-Nikodym-Eigenschaft">Radon-Nikodym-Eigenschaft</a></div>

Der <a href="Satz_von_Radon-Nikod%C3%BDm" title="Satz von Radon-Nikodým">Satz von Radon-Nikodým</a> gilt für das Bochner-Integral im Allgemeinen nicht. Banachräume, für die dieser Satz gilt, bezeichnet man als Banachräume mit der Radon-Nikodym-Eigenschaft. <a href="Reflexiver_Raum" title="Reflexiver Raum">Reflexive Räume</a> besitzen stets die Radon-Nikodym-Eigenschaft.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Bochner-Lebesgue-Räume">Bochner-Lebesgue-Räume</h2></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}},\mu )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cabc7dd1ffb04becc182e5356682ec027db7211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},\mu )}" loading="lazy"> ein <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">-endlicher, vollständiger Maßraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B,\|\cdot \|)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d8873f311a155a76f4abcef9649478b53911372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.611ex; height:2.843ex;" alt="{\displaystyle (B,\|\cdot \|)}" loading="lazy"> ein Banachraum, so nennt man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }" loading="lazy"> den Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,{\mathcal {A}},\mu ,B)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c30cb0fe73341ee94750f96bf88245ab5188a17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.3ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,{\mathcal {A}},\mu ,B)}" loading="lazy"> der Bochner-integrierbaren Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \rightarrow B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9408f1d3124d3ab47804661cac07ccae3102551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.056ex; height:2.176ex;" alt="{\displaystyle \Omega \rightarrow B}" loading="lazy"> einen Bochner-Lebesgue-Raum, wobei wie üblich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-fast gleiche Funktionen identifiziert werden durch <a href="%C3%84quivalenzklasse" class="mw-redirect" title="Äquivalenzklasse">Äquivalenzklassen</a>. Man erhält mit der Norm

<msub>

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</msub>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }\|f(\omega )\|^{p}\mathrm {d} \mu (\omega )\right)^{1/p},\quad 1\leq p<\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ccbded863519da8ce5c6c1320ed2730b28352f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.707ex; height:6.676ex;" alt="{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }\|f(\omega )\|^{p}\mathrm {d} \mu (\omega )\right)^{1/p},\quad 1\leq p<\infty }" loading="lazy"></dd>

<msub>

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</msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }:=\mathrm {ess} \sup \|f(\omega )\|,\quad \quad p=\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ade917027d577c4dc954adfad76e85510f09838b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.494ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }:=\mathrm {ess} \sup \|f(\omega )\|,\quad \quad p=\infty }" loading="lazy"></dd></dl>

einen Banachraum. Dieser lässt sich für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c29a2f2fb3f642618036ed7a79712202e7ada924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}" loading="lazy"> wie folgt als <a href="Tensorprodukt" title="Tensorprodukt">Tensorprodukt</a> beschreiben.

Man rechnet nach, dass durch

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\times B\rightarrow L^{1}(\Omega ,{\mathcal {A}},\mu ,B),\,(f,\alpha )\mapsto f(\cdot )\alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a494315839063a2da0377e7626d829f7def16e67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.878ex; height:3.176ex;" alt="{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\times B\rightarrow L^{1}(\Omega ,{\mathcal {A}},\mu ,B),\,(f,\alpha )\mapsto f(\cdot )\alpha }" loading="lazy"></dd></dl>

eine <a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilineare Abbildung</a> gegeben ist, die einen isometrischen Isomorphismus

<msup>

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</msup>

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<msub>

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\mathbin {{\hat {\otimes }}_{\pi }} B\cong L^{1}(\Omega ,{\mathcal {A}},\mu ,B)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd5b015c79c8eab4ac56e48543f5f7c424d9f42e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.67ex; height:3.176ex;" alt="{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\mathbin {{\hat {\otimes }}_{\pi }} B\cong L^{1}(\Omega ,{\mathcal {A}},\mu ,B)}" loading="lazy"></dd></dl>

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

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ff5a5fca23838f70344828a4b0e3b8ee349307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.843ex;" alt="{\displaystyle {\hat {\otimes }}_{\pi }}" loading="lazy"> das <a href="Projektives_Tensorprodukt" title="Projektives Tensorprodukt">projektive Tensorprodukt</a> bezeichne.<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Erweiterung_auf_lokalkonvexe_Räume">Erweiterung auf lokalkonvexe Räume</h2></div>

Es ist möglich das Bochner-Integral auf <a href="Lokalkonvexer_Raum" title="Lokalkonvexer Raum">lokalkonvexe Räume</a> zu erweitern, dies wurde 1975 von Wjatscheslaw Rybakow<a href="#cite_note-9">[9]</a>, 1981 durch Chris Blondia<a href="#cite_note-10">[10]</a> und 2015 von Ralf Beckmann und <a href="Anton_Deitmar" title="Anton Deitmar">Anton Deitmar</a> gemacht<a href="#cite_note-11">[11]</a>, wobei Beckmann und Deitmar den ursprünglichen Ansatz von Bochner für vektorwertige Integrale auf <a href="Netz_(Topologie)" title="Netz (Topologie)">Netze</a> erweiterten.

Die nachfolgende Definition stammt von Blondia:<a href="#cite_note-12">[12]</a>

Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}},\mu )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cabc7dd1ffb04becc182e5356682ec027db7211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}},\mu )}" loading="lazy"> ein <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">-endlicher, <a href="Vollst%C3%A4ndiger_Ma%C3%9Fraum" class="mw-redirect" title="Vollständiger Maßraum">vollständiger Maßraum</a>. Weiter sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {P}})}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99067ca32350a246852d46e67c855905fc8ef57d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.527ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {P}})}" loading="lazy"> ein hausdorffscher lokalkonvexer Raum, der vollständig ist und dessen Topologie durch eine Familie von <a href="Seminorm" class="mw-redirect" title="Seminorm">Seminormen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"> erzeugt wird. Eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \to X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb19ff7d838c8ed65ae679b65862ddef67c69bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.585ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \to X}" loading="lazy"> heißt stark-integrierbar oder Bochner-integrierbar, wenn eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a32c571f35fd41f5ef78fecc130ac39cd2108c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.167ex; height:2.843ex;" alt="{\displaystyle (f_{n})}" loading="lazy"> existiert, so dass

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f_{n}(\omega )\to f(\omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f916e6e6cf42c1da3ab32d935d1f9170acaf9918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.761ex; height:2.843ex;" alt="{\displaystyle f_{n}(\omega )\to f(\omega )}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">-fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }">

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</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }" loading="lazy"> gilt.</li>

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<annotation encoding="application/x-tex">{\displaystyle p(f(\omega )-f_{n}(\omega ))\in L^{1}(\Omega ;\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/085152d7b8e88268a81bb4410ec8400251f04676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:27.732ex; height:3.176ex;" alt="{\displaystyle p(f(\omega )-f_{n}(\omega ))\in L^{1}(\Omega ;\mathbb {R} )}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> und alle Seminormen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in {\mathcal {P}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7e2c46d3c723d2e8ccc606a4d3333927e6b375e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.803ex; height:2.509ex;" alt="{\displaystyle p\in {\mathcal {P}}}" loading="lazy">, das heißt</li></ul>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \lim \limits _{n\to \infty }\int _{\Omega }p(f(\omega )-f_{n}(\omega ))d\mu =0.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e5ff86ec06874b0254bd61dd8619a1e308f7d1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.249ex; height:5.676ex;" alt="{\displaystyle \lim \limits _{n\to \infty }\int _{\Omega }p(f(\omega )-f_{n}(\omega ))d\mu =0.}" loading="lazy"></dd></dl></dd></dl>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int _{A}f_{n}(\omega )d\mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41b64f0592afaae6e986ca4dc794d8d114f7d61e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.375ex; height:5.676ex;" alt="{\displaystyle \int _{A}f_{n}(\omega )d\mu }" loading="lazy"> konvergiert für jede messbare Teilmenge <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"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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

</semantics>

</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">.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Ansatz_von_Beckmann_und_Deitmar">Ansatz von Beckmann und Deitmar</h3></div>

Beckmann und Deitmar verwenden den Begriff der Bochner-Approximierbarkeit für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \Omega \to X}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb19ff7d838c8ed65ae679b65862ddef67c69bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.585ex; height:2.509ex;" alt="{\displaystyle f\colon \Omega \to X}" loading="lazy"> als Voraussetzung für die Bochner-Integrierbarkeit und geben eine Charakterisierung dieses Begriffs. Eine Funktion heißt Bochner-approximierbar falls ein Netz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s_{j})_{j\in J}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (s_{j})_{j\in J}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/696929740bec7d6bb4d700737f4c747928b2bff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.856ex; height:3.009ex;" alt="{\displaystyle (s_{j})_{j\in J}}" loading="lazy"> von einfachen Funktionen existiert, so dass für jede stetige Seminorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> auf <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_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.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">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }p(f-s_{j})d\mu \to 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aabcb0b0bbabe4c34e305395eec36ae39a157bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.59ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }p(f-s_{j})d\mu \to 0}" loading="lazy"></dd></dl>

gilt. Eine alternative Bedingung ohne den Begriff des Netzes lautet wie folgt: <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"> ist Bochner-approximierbar, falls für jede stetige Seminorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> eine einfache Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/577e6806693dcbbd731632cb9e9ead8d66947cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.15ex; height:2.343ex;" alt="{\displaystyle s_{p}}" loading="lazy"> mit der Eigenschaft

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }p(f-s_{p})d\mu <1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b276c4c457085987a02f6aeb506c9276ba2336b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.224ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }p(f-s_{p})d\mu <1}" loading="lazy"></dd></dl>

existiert.

Sie unterscheiden zwischen drei Fällen an Anforderungen an den lokalkonvexen Raum <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_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.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"><a href="#cite_note-13">[13]</a>

</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"> ist vollständig</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"> ist <a href="Vollst%C3%A4ndiger_Raum#Quasivollständiger_Raum" title="Vollständiger Raum">quasivollständig</a> und die 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"> ist beschränkt,</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"> ist quasivollständig und das Maß ist endlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (\Omega )<\infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/885316f6322dd3829578fd2ae8572e6be0b4f9d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.311ex; height:2.843ex;" alt="{\displaystyle \mu (\Omega )<\infty }" loading="lazy">.</li></ul>

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

<ul><li><a href="Birkhoff-Integral" title="Birkhoff-Integral">Birkhoff-Integral</a></li>

<li><a href="Pettis-Integral" title="Pettis-Integral">Pettis-Integral</a></li></ul>

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

<ul><li>Herbert Amann, <a href="Joachim_Escher_(Mathematiker)" title="Joachim Escher (Mathematiker)">Joachim Escher</a>: Analysis. Band 3. Birkhäuser, Basel u. a. 2001, ISBN 3-7643-6613-3.</li>

<li>Malempati M. Rao: Measure Theory and Integration (= Pure and Applied Mathematics. A Program of Monographs, Textbooks, and Lecture Notes. Bd. 265). 2nd edition, revised and expanded. Dekker, New York NY u. a. 2004, ISBN 0-8247-5401-8, S. 505 ff.</li></ul>

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

<ul><li><a href="Salomon_Bochner" title="Salomon Bochner">Salomon Bochner</a>: <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm20/fm20127.pdf">Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind.</a> (PDF; 799 kB). In: <a href="Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a>. Bd. 20, 1933, S. 262–276.</li>

<li>V. I. Sobolev: <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Bochner_integral">Bochner integral.</a> In: <a href="Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a> (englisch).</li>

<li><a rel="nofollow" class="external text" href="http://www.matheplanet.com/matheplanet/nuke/html/article.php?sid=1101">Integrale vektorwertiger Funktionen.</a> In: <a href="Matroids_Matheplanet" title="Matroids Matheplanet">Matroids Matheplanet</a>.</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Herbert Amann, Joachim Escher: Analysis. Band 3. 2001, Bemerkung X.2.1 (a).

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Herbert Amann, Joachim Escher: Analysis. Band 3. 2001, S. 65.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Herbert Amann, Joachim Escher: Analysis. Band 3. 2001, S. 87.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Herbert Amann, Joachim Escher: Analysis. Band 3. 2001, Korollar X.2.7.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Herbert Amann, Joachim Escher: Analysis. Band 3. 2001, S. 94.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Herbert Amann, Joachim Escher: Analysis. Band 3. 2001, S. 92.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Joseph Diestel, John Jerry Uhl: Vector Measures (= Mathematical Surveys. Bd. 15). American Mathematical Society, Providence RI 1977, ISBN 0-8218-1515-6, Corollary III.2.13.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer-Verlag 2002, ISBN 1-85233-437-1, Beispiel 2.19

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Vyacheslav I. Rybakov: <cite style="font-style:italic">A generalization of the Bochner integral to locally convex spaces</cite>. In: <cite style="font-style:italic">Mathematical Notes of the Academy of Sciences of the USSR</cite>. Band 18, 1975, S. 933–938, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01153047">10.1007/BF01153047</a>.<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:Bochner-Integral&rft.atitle=A+generalization+of+the+Bochner+integral+to+locally+convex+spaces&rft.au=Vyacheslav+I.+Rybakov&rft.btitle=Mathematical+Notes+of+the+Academy+of+Sciences+of+the+USSR&rft.date=1975&rft.doi=10.1007%2FBF01153047&rft.genre=book&rft.pages=933-938&rft.volume=18" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Chris Blondia: <cite style="font-style:italic">Integration in locally convex spaces</cite>. In: <cite style="font-style:italic">Simon Stevin, (A Quarterly Journal of Pure and Applied Math.)</cite>. Band 55, Nr. 3, 1981, S. 81–102.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bochner-Integral&rft.atitle=Integration+in+locally+convex+spaces&rft.au=Chris+Blondia&rft.date=1981&rft.genre=journal&rft.issue=3&rft.jtitle=Simon+Stevin%2C+%28A+Quarterly+Journal+of+Pure+and+Applied+Math.%29&rft.pages=81-102&rft.volume=55" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Ralf Beckmann und <a href="Anton_Deitmar" title="Anton Deitmar">Anton Deitmar</a>: <cite style="font-style:italic">Two applications of nets</cite>. In: <cite style="font-style:italic">Ann. Funct. Anal.</cite> Band 6, Nr. 3, 2015, S. 176 - 190, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.15352/afa%2F06-3-15">10.15352/afa/06-3-15</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bochner-Integral&rft.atitle=Two+applications+of+nets&rft.au=Ralf+Beckmann+und+Anton+Deitmar&rft.date=2015&rft.doi=10.15352%2Fafa%2F06-3-15&rft.genre=journal&rft.issue=3&rft.jtitle=Ann.+Funct.+Anal.&rft.pages=176+-+190&rft.volume=6" style="display:none">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Valeria Marraffa: <cite style="font-style:italic">A Birkhoff Type Integral and the Bourgain Property in a Locally Convex Space</cite>. In: Michigan State University Press (Hrsg.): <cite style="font-style:italic">Real Analysis Exchange</cite>. Band 32, Nr. 2, 2006, S. 410 (<a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/real-analysis-exchange/volume-32/issue-2/A-Birkhoff-Type-Integral-and-the-Bourgain-Property-in-a/rae/1199377481.full">projecteuclid.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bochner-Integral&rft.atitle=A+Birkhoff+Type+Integral+and+the+Bourgain+Property+in+a+Locally+Convex+Space&rft.au=Valeria+Marraffa&rft.date=2006&rft.genre=journal&rft.issue=2&rft.jtitle=Real+Analysis+Exchange&rft.pages=410&rft.volume=32" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Ralf Beckmann und <a href="Anton_Deitmar" title="Anton Deitmar">Anton Deitmar</a>: <cite style="font-style:italic">Two applications of nets</cite>. In: <cite style="font-style:italic">Ann. Funct. Anal.</cite> Band 6, Nr. 3, 2015, S. 183, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.15352/afa%2F06-3-15">10.15352/afa/06-3-15</a> (Theorem 1.15 und Theorem 1.16).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bochner-Integral&rft.atitle=Two+applications+of+nets&rft.au=Ralf+Beckmann+und+Anton+Deitmar&rft.date=2015&rft.doi=10.15352%2Fafa%2F06-3-15&rft.genre=journal&rft.issue=3&rft.jtitle=Ann.+Funct.+Anal.&rft.pages=183&rft.volume=6" style="display:none">

</li>

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