GermanWikipedia - /page/entry.mu

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<title>Pettis-Integral</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Pettis-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 Pettis-Integral ist ein nach <a href="Billy_James_Pettis" title="Billy James Pettis">Billy James Pettis</a> benannter Begriff aus dem mathematischen Teilgebiet der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>. Es handelt sich um ein Integral für Funktionen auf einem <a href="Ma%C3%9Fraum" title="Maßraum">Maßraum</a> mit Werten in einem <a href="Banachraum" title="Banachraum">Banachraum</a>. Ist der Banachraum gleich dem eindimensionalen Raum <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">, so erhält man das übliche Integral reellwertiger Funktionen auf dem Maßraum. Das Pettis-Integral verallgemeinert aber nicht nur das Integral reellwertiger Funktionen, sondern auch das <a href="Bochner-Integral" title="Bochner-Integral">Bochner-Integral</a> und das <a href="Birkhoff-Integral" title="Birkhoff-Integral">Birkhoff-Integral</a>, welche ebenfalls Integrale Banachraum-wertiger Funktionen sind.

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

Wir gehen von einem <a href="Vollst%C3%A4ndiger_Ma%C3%9Fraum" class="mw-redirect" title="Vollständiger Maßraum">vollständigen Maßraum</a> <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"> mit einem endlichen, positiven <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Maß</a> <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"> aus und wollen für Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> mit Werten in einem Banachraum <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"> ein Integral definieren. Für die im Folgenden beschriebene Konstruktion nutzen wir aus, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48bc0467914ba2ece3280bc232bea7570f4043d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.993ex; height:2.676ex;" alt="{\displaystyle \varphi \circ f}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> aus dem <a href="Dualraum" title="Dualraum">Dualraum</a> <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_/eb734a37dd21ce173a46342d1cc64c92/865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"> eine reellwertige Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \rightarrow \mathbb {R} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b8f706baed54c3c860d22e3ef2538ffca32542f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle \Omega \rightarrow \mathbb {R} }" loading="lazy"> ist und dass <a href="Ma%C3%9Ftheorie" title="Maßtheorie">maßtheoretische</a> Begriffe für solche Funktionen bereits definiert sind. Wir nennen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> schwach-messbar, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48bc0467914ba2ece3280bc232bea7570f4043d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.993ex; height:2.676ex;" alt="{\displaystyle \varphi \circ f}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in X'}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa41a7bbaba1963c659b4510fff62aec38de3a22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.042ex; height:3.009ex;" alt="{\displaystyle \varphi \in X'}" loading="lazy"> eine <a href="Messbare_Funktion" title="Messbare Funktion">messbare Funktion</a> ist. Dagegen nennt man <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"> wie üblich messbar, wenn das Urbild jeder offenen Menge aus <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"> ist. Für die Beziehung dieser beiden Messbarkeitsbegriffe siehe den <a href="Bochner-Integral#Messbarkeitssatz_von_Pettis" title="Bochner-Integral">Messbarkeitssatz von Pettis</a>. Schließlich nennen wir <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"> schwach-integrierbar, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48bc0467914ba2ece3280bc232bea7570f4043d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.993ex; height:2.676ex;" alt="{\displaystyle \varphi \circ f}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in X'}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa41a7bbaba1963c659b4510fff62aec38de3a22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.042ex; height:3.009ex;" alt="{\displaystyle \varphi \in X'}" loading="lazy"> eine <a href="Integrierbare_Funktion" class="mw-redirect" title="Integrierbare Funktion">integrierbare Funktion</a> ist.

Wir betrachten nun eine schwach-integrierbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy">. Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in X'}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa41a7bbaba1963c659b4510fff62aec38de3a22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.042ex; height:3.009ex;" alt="{\displaystyle \varphi \in X'}" loading="lazy"> ist dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ f\in L^{1}(\Omega ,{\mathcal {A}},\mu )}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \circ f\in L^{1}(\Omega ,{\mathcal {A}},\mu )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b81e5187be59afb28339e5d048cad29c46c65df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.331ex; height:3.176ex;" alt="{\displaystyle \varphi \circ f\in L^{1}(\Omega ,{\mathcal {A}},\mu )}" loading="lazy">, wobei letzteres den <a href="Lp-Raum" title="Lp-Raum">L1-Raum</a> über dem vorgegebenen Maßraum bezeichne, der nach dem <a href="Satz_von_Fischer-Riesz" title="Satz von Fischer-Riesz">Satz von Fischer-Riesz</a> bzgl. der <a href="P-Norm" title="P-Norm">1-Norm</a> ein Banachraum ist. Wir erhalten damit einen <a href="Linearer_Operator" title="Linearer Operator">linearen Operator</a>

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<annotation encoding="application/x-tex">{\displaystyle S_{f}:X'\rightarrow L^{1}(\Omega ,{\mathcal {A}},\mu ),\,\varphi \mapsto \varphi \circ f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/476d05020f9a893912be9110c10f0522b9f500da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.84ex; height:3.343ex;" alt="{\displaystyle S_{f}:X'\rightarrow L^{1}(\Omega ,{\mathcal {A}},\mu ),\,\varphi \mapsto \varphi \circ f}" loading="lazy">,</dd></dl>

von dem man mittels des <a href="Satz_vom_abgeschlossenen_Graphen" title="Satz vom abgeschlossenen Graphen">Satzes vom abgeschlossenen Graphen</a> zeigen kann, dass er sogar <a href="Beschr%C3%A4nkter_Operator" title="Beschränkter Operator">stetig</a> ist.

Man kann daher den <a href="Adjungierter_Operator#Verallgemeinerung_auf_Banachräume" title="Adjungierter Operator">adjungierten Operator</a>

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<annotation encoding="application/x-tex">{\displaystyle T_{f}=S_{f}':L^{1}(\Omega ,{\mathcal {A}},\mu )'\rightarrow X''}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5967a7a9d432c0e155ef49cd84e596db57ecd114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.021ex; height:3.676ex;" alt="{\displaystyle T_{f}=S_{f}':L^{1}(\Omega ,{\mathcal {A}},\mu )'\rightarrow X''}" loading="lazy"> bilden. Identifiziert man den Dualraum von L1 mittels <a href="Dualit%C3%A4t_von_Lp-R%C3%A4umen" title="Dualität von Lp-Räumen">Lp-Dualität</a> wie üblich mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\Omega ,{\mathcal {A}},\mu )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b7099810c55a746bc980b0c59953d827287681d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.319ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\Omega ,{\mathcal {A}},\mu )}" loading="lazy">, so erhält man einen Operator

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<annotation encoding="application/x-tex">{\displaystyle T_{f}:L^{\infty }(\Omega ,{\mathcal {A}},\mu )\rightarrow X'',\quad (T_{f}(g))(\varphi )=\int _{\Omega }g(t)\varphi (f(t))\,\mathrm {d} \mu (t),\quad \quad g\in L^{\infty }(\Omega ,{\mathcal {A}},\mu ),\varphi \in X'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c2c30783d97820717f2af1ef078f2fffb5d2f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:90.392ex; height:5.676ex;" alt="{\displaystyle T_{f}:L^{\infty }(\Omega ,{\mathcal {A}},\mu )\rightarrow X'',\quad (T_{f}(g))(\varphi )=\int _{\Omega }g(t)\varphi (f(t))\,\mathrm {d} \mu (t),\quad \quad g\in L^{\infty }(\Omega ,{\mathcal {A}},\mu ),\varphi \in X'}" loading="lazy">.</dd></dl>

Insbesondere kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{f}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbb8cb970fa8b6a14c3edcbd6951437428003b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.494ex; height:2.843ex;" alt="{\displaystyle T_{f}}" loading="lazy"> auf <a href="Indikatorfunktion" title="Indikatorfunktion">charakteristische Funktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{E}:\Omega \rightarrow \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{E}:\Omega \rightarrow \{0,1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/404c3ba46c64dcc44e411b4f32838f81bee83015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.856ex; height:2.843ex;" alt="{\displaystyle \chi _{E}:\Omega \rightarrow \{0,1\}}" loading="lazy"> für messbare Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in {\mathcal {A}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617e7bf2f419fd8a31234ec2b0d5c9df00457025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.343ex;" alt="{\displaystyle E\in {\mathcal {A}}}" loading="lazy"> anwenden. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{f}(\chi _{E})\in X''}">

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<annotation encoding="application/x-tex">{\displaystyle T_{f}(\chi _{E})\in X''}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4281a4e0410a68df4a0130c6e53dff184f689b6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.221ex; height:3.176ex;" alt="{\displaystyle T_{f}(\chi _{E})\in X''}" loading="lazy"> nennt man das Dunford-Integral<a href="#cite_note-1">[1]</a>, nach <a href="Nelson_Dunford" title="Nelson Dunford">Nelson Dunford</a>, oder das Gelfand-Integral<a href="#cite_note-2">[2]</a>, nach <a href="Israel_Gelfand" class="mw-redirect" title="Israel Gelfand">Israel Gelfand</a>, und schreibt

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<annotation encoding="application/x-tex">{\displaystyle \int _{E}f\,\mathrm {d} \mu =\int _{E}f(t)\,\mathrm {d} \mu (t):=T_{f}(\chi _{E})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46888f656361bb6833c97efdcfdfe316eac0e09e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.442ex; height:5.676ex;" alt="{\displaystyle \int _{E}f\,\mathrm {d} \mu =\int _{E}f(t)\,\mathrm {d} \mu (t):=T_{f}(\chi _{E})}" loading="lazy">.</dd></dl>

Stellt man sich ein Integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0865a4d16c5132329cd9ba16036ad7f734e5e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.332ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu }" loading="lazy"> einer Funktion mit Werten in <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"> als <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">-Mittelung der <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">-Werte vor, so wird man erwarten, dass das Integral wieder in <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"> liegt. Im Allgemeinen ist das nicht der Fall. Nun ist aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset X''}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6d97dead6c4b29518ef75b4cff53f312e201fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.212ex; height:2.509ex;" alt="{\displaystyle X\subset X''}" loading="lazy"> durch die sogenannte <a href="Dualraum#Bidualraum" title="Dualraum">kanonische Einbettung</a>, daher definiert man:

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> heißt Pettis-integrierbar, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu \in X}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu \in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2674dfb062fa52390476504b33d8736e68b39a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.153ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu \in X}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617e7bf2f419fd8a31234ec2b0d5c9df00457025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.343ex;" alt="{\displaystyle E\in {\mathcal {A}}}" loading="lazy">, und man nennt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0865a4d16c5132329cd9ba16036ad7f734e5e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.332ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu }" loading="lazy"> das Pettis-Integral von <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"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">.

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

<div class="mw-heading mw-heading3"><h3 id="Reflexive_Räume">Reflexive Räume</h3></div>

Ist <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="Reflexiver_Raum" title="Reflexiver Raum">reflexiv</a>, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=X''}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d6892fc6c59841595f3e0a3b981a27dd0dea04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.212ex; height:2.509ex;" alt="{\displaystyle X=X''}" loading="lazy"> und es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu \in X}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu \in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2674dfb062fa52390476504b33d8736e68b39a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.153ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{E}f\,\mathrm {d} \mu \in X}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617e7bf2f419fd8a31234ec2b0d5c9df00457025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.343ex;" alt="{\displaystyle E\in {\mathcal {A}}}" loading="lazy"> und jede schwach-integrierbare 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">. Das heißt, dass jede schwach-integrierbare Funktion mit Werten in einem reflexiven Raum Pettis-integrierbar ist.

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

Jede <a href="Birkhoff-Integral" title="Birkhoff-Integral">Birkhoff-integrierbare</a> Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> ist Pettis-integrierbar und das Birkhoff-Integral stimmt mit dem Pettis-Integral überein. Daher ist das Pettis-Integral eine Verallgemeinerung des Birkhoff-Integrals.

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

Jede <a href="Bochner-Integral" title="Bochner-Integral">Bochner-integrierbare</a> Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> ist Pettis-integrierbar und das Bochner-Integral stimmt mit dem Pettis-Integral überein. Deshalb ist das Pettis-Integral auch eine Verallgemeinerung des Bochner-Integrals. Es gilt

<dl><dd>Bochner-integrierbar   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy">   Birkhoff-integrierbar   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy">   Pettis-integrierbar   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy">   schwach-integrierbar.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Pettis-integrierbar_aber_nicht_Bochner-integrierbar">Pettis-integrierbar aber nicht Bochner-integrierbar</h3></div>

Als Maßraum betrachten wir das Einheitsintervall [0,1] mit dem <a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/588a981eb3c6f32c01153f8710a7f70029b5e553.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} t}" loading="lazy"> auf der <a href="Sigma-Algebra" class="mw-redirect" title="Sigma-Algebra">σ-Algebra</a> der Lebesgue-messbaren Mengen und als Banachraum den <a href="Folgenraum" title="Folgenraum">Folgenraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/480410f0d709ad324aa9b82c60619402a9918fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.14ex; height:2.509ex;" alt="{\displaystyle X=c_{0}}" loading="lazy"> der reellen <a href="Nullfolge" title="Nullfolge">Nullfolgen</a>. Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aba34f081d776e30204f3458e4f50b403b09e5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}" loading="lazy"> das <a href="Halboffenes_Intervall" class="mw-redirect" title="Halboffenes Intervall">halboffene Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle ({\frac {1}{n+1}},{\frac {1}{n}}]\subset [0,1]}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle ({\frac {1}{n+1}},{\frac {1}{n}}]\subset [0,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ec8a8a8804c7ba8d3e6900e4af067c53f84eea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.082ex; height:3.676ex;" alt="{\displaystyle \textstyle ({\frac {1}{n+1}},{\frac {1}{n}}]\subset [0,1]}" loading="lazy"> und

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:[0,1]\rightarrow c_{0},\,f(t):=(n\chi _{I_{n}}(t))_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/752b8f84e2406397273ae5156051fa101b72f05d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.367ex; height:2.843ex;" alt="{\displaystyle f:[0,1]\rightarrow c_{0},\,f(t):=(n\chi _{I_{n}}(t))_{n\in \mathbb {N} }}" loading="lazy">.</dd></dl>

Jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"> ist tatsächlich eine Nullfolge. Das ist klar für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">, denn es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=(0,0,0,\ldots )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=(0,0,0,\ldots )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83624d3e182e17f063dc4e721a8b52b1eea5f1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.471ex; height:2.843ex;" alt="{\displaystyle f(0)=(0,0,0,\ldots )}" loading="lazy"> und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29a2960e88369263fe3cfe00ccbfeb83daee212a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}" loading="lazy"> gibt es genau ein <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"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in I_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7e05c8c3183fc6e72c6ba90373ee35d35c8aea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.922ex; height:2.509ex;" alt="{\displaystyle t\in I_{n}}" loading="lazy"> und daher ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=(0,0,\ldots ,n,0,0,\ldots )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)=(0,0,\ldots ,n,0,0,\ldots )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4dc9fe12479009b0e0cb87bf8c356d1084ca828.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.917ex; height:2.843ex;" alt="{\displaystyle f(t)=(0,0,\ldots ,n,0,0,\ldots )}" loading="lazy">. Diese Funktion ist Pettis-integrierbar aber nicht Bochner-integrierbar. Zur Verdeutlichung obiger Konstruktionen führen wir die erforderlichen Rechnungen aus und beginnen mit der schwachen Integrierbarkeit.

Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3be1917aa8a9460a761f248e7cb10058b06db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.017ex; height:3.343ex;" alt="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}" loading="lazy"> ist nach Definition der <a href="Dualraum" title="Dualraum">Dualität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}'=\ell ^{1}}">

</mrow>

</msubsup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle c_{0}'=\ell ^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c41397882fc4ca88570076f9459099cd0febee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.183ex; height:3.343ex;" alt="{\displaystyle c_{0}'=\ell ^{1}}" loading="lazy">

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</munderover>

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<msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (\varphi \circ f)(t)=\sum _{n=1}^{\infty }v_{n}\cdot n\chi _{I_{n}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12cde5c2fa4112710454ebc2ad66699781a5e086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.737ex; height:6.843ex;" alt="{\displaystyle (\varphi \circ f)(t)=\sum _{n=1}^{\infty }v_{n}\cdot n\chi _{I_{n}}(t)}" loading="lazy"></dd></dl>

und daher

<msub>

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<msub>

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<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{[0,1]}|(\varphi \circ f)(t)|\,\mathrm {d} t\leq \sum _{n=1}^{\infty }|v_{n}|\cdot n\int _{[0,1]}\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }|v_{n}|\cdot n{\frac {1}{n(n+1)}}=\sum _{n=1}^{\infty }|v_{n}|\cdot {\frac {1}{n+1}}<\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9a91aff3814d5466f026a89f06c843ab7871575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:92.413ex; height:6.843ex;" alt="{\displaystyle \int _{[0,1]}|(\varphi \circ f)(t)|\,\mathrm {d} t\leq \sum _{n=1}^{\infty }|v_{n}|\cdot n\int _{[0,1]}\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }|v_{n}|\cdot n{\frac {1}{n(n+1)}}=\sum _{n=1}^{\infty }|v_{n}|\cdot {\frac {1}{n+1}}<\infty }" loading="lazy">,</dd></dl>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aba34f081d776e30204f3458e4f50b403b09e5c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}" loading="lazy"> hat die Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{n(n+1)}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{n(n+1)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8402d9f5853818b1f09d0ac658ce55da2d15d477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {1}{n(n+1)}}}" loading="lazy">. Also ist <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"> schwach-integrierbar.

Zur Bestimmung der Gelfand-Integrale betrachte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in L^{\infty }([0,1])}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in L^{\infty }([0,1])}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f586a87cdccc625d57bd87909682c5a1d541747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.877ex; height:2.843ex;" alt="{\displaystyle g\in L^{\infty }([0,1])}" loading="lazy">. Bezeichnen wir die L1-L∞-Dualität mit spitzen Klammern, so ist für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}">

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3be1917aa8a9460a761f248e7cb10058b06db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.017ex; height:3.343ex;" alt="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}" loading="lazy">

<msub>

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<msub>

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</mrow>

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</munderover>

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<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle T_{f}(g),\varphi \rangle =\langle g,S_{f}(\varphi )\rangle =\langle g,\varphi \circ f\rangle =\int _{[0,1]}g(t)(\varphi \circ f)(t)\,\mathrm {d} t=\int _{[0,1]}g(t)\sum _{n=1}^{\infty }v_{n}\cdot n\chi _{I_{n}}(t)\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83f31aed361f14a7e5800dbfadb9ffc2f07396e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:88.429ex; height:6.843ex;" alt="{\displaystyle \langle T_{f}(g),\varphi \rangle =\langle g,S_{f}(\varphi )\rangle =\langle g,\varphi \circ f\rangle =\int _{[0,1]}g(t)(\varphi \circ f)(t)\,\mathrm {d} t=\int _{[0,1]}g(t)\sum _{n=1}^{\infty }v_{n}\cdot n\chi _{I_{n}}(t)\,\mathrm {d} t}" loading="lazy"></dd>

</mrow>

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</munderover>

<msub>

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<msub>

<msub>

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</munderover>

<msub>

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<msub>

<msub>

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</mrow>

<msub>

<mrow>

<mrow>

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<msub>

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</mrow>

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</mrow>

</mrow>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mrow>

<mrow>

<msub>

<msub>

</mrow>

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</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\sum _{n=1}^{\infty }v_{n}\cdot n\int _{[0,1]}g(t)\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }v_{n}\cdot n\int _{I_{n}}g(t)\,\mathrm {d} t=\langle \left(n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right)_{n\in \mathbb {N} },(v_{n})_{n\in \mathbb {N} }\rangle =\langle \left(n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right)_{n\in \mathbb {N} },\varphi \rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f0160b052616a8b9c0c32922219d364b3373899.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:112.593ex; height:6.843ex;" alt="{\displaystyle =\sum _{n=1}^{\infty }v_{n}\cdot n\int _{[0,1]}g(t)\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }v_{n}\cdot n\int _{I_{n}}g(t)\,\mathrm {d} t=\langle \left(n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right)_{n\in \mathbb {N} },(v_{n})_{n\in \mathbb {N} }\rangle =\langle \left(n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right)_{n\in \mathbb {N} },\varphi \rangle }" loading="lazy"></dd></dl>

und man liest ab

<msub>

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<annotation encoding="application/x-tex">{\displaystyle T_{f}(g)=\left(n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right)_{n\in N}\in \ell ^{\infty }=c_{0}''}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30f822f1f335286c8840d45051434d4b88a16aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.599ex; height:6.343ex;" alt="{\displaystyle T_{f}(g)=\left(n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right)_{n\in N}\in \ell ^{\infty }=c_{0}''}" loading="lazy">.</dd></dl>

Tatsächlich liegt diese Folge aber bereits in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy">, denn

<mrow>

<mrow>

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<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right|\leq \|g\|_{\infty }n\cdot \int _{I_{n}}\mathrm {d} t=\|g\|_{\infty }\cdot {\frac {1}{n+1}}\rightarrow 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64becb2a8d410052f3f6e80fe26113f4aae3ae6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.577ex; height:6.509ex;" alt="{\displaystyle \left|n\cdot \int _{I_{n}}g(t)\,\mathrm {d} t\right|\leq \|g\|_{\infty }n\cdot \int _{I_{n}}\mathrm {d} t=\|g\|_{\infty }\cdot {\frac {1}{n+1}}\rightarrow 0}" loading="lazy">.</dd></dl>

Daher ist <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"> Pettis-integrierbar. <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 aber nicht Bochner-integrierbar, denn

<msub>

<msub>

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</mrow>

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</mrow>

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</munderover>

<msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\mapsto \|f(t)\|=\|(n\chi _{I_{n}}(t)\|=\sum _{n=1}^{\infty }n\chi _{I_{n}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1403a869fb23e2feed706ad6c5f6c11d29727980.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.713ex; height:6.843ex;" alt="{\displaystyle t\mapsto \|f(t)\|=\|(n\chi _{I_{n}}(t)\|=\sum _{n=1}^{\infty }n\chi _{I_{n}}(t)}" loading="lazy"></dd></dl>

ist nicht integrierbar.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Schwach-integrierbar_aber_nicht_Pettis-integrierbar">Schwach-integrierbar aber nicht Pettis-integrierbar</h3></div>

Zur Konstruktion einer schwach-integrierbaren Funktion, die nicht Pettis-integrierbar ist, wandeln wir obiges Beispiel leicht ab.

Wieder betrachten wir den Maßraum [0,1] mit dem <a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a> und den Banachraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

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</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy">. Die gesuchte Funktion ist

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<annotation encoding="application/x-tex">{\displaystyle f:[0,1]\rightarrow c_{0},\,f(t):=(n(n+1)\chi _{I_{n}}(t))_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2ca035a9f927e29f76dacffb2f8c3c8eef612dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.574ex; height:2.843ex;" alt="{\displaystyle f:[0,1]\rightarrow c_{0},\,f(t):=(n(n+1)\chi _{I_{n}}(t))_{n\in \mathbb {N} }}" loading="lazy">.</dd></dl>

Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3be1917aa8a9460a761f248e7cb10058b06db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.017ex; height:3.343ex;" alt="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}" loading="lazy"> ist

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<annotation encoding="application/x-tex">{\displaystyle (\varphi \circ f)(t)=\sum _{n=1}^{\infty }v_{n}\cdot n(n+1)\chi _{I_{n}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/569e15567a392de5ec616c3f19ebe4ce6f13bfa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.944ex; height:6.843ex;" alt="{\displaystyle (\varphi \circ f)(t)=\sum _{n=1}^{\infty }v_{n}\cdot n(n+1)\chi _{I_{n}}(t)}" loading="lazy"></dd></dl>

und daher

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \int _{[0,1]}|(\varphi \circ f)(t)|\,\mathrm {d} t\leq \sum _{n=1}^{\infty }|v_{n}|\cdot n(n+1)\int _{[0,1]}\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }|v_{n}|\cdot n(n+1){\frac {1}{n(n+1)}}=\sum _{n=1}^{\infty }|v_{n}|<\infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/550ff076e462ed71ab647c5dc46fcf38b26a2aef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:98.914ex; height:6.843ex;" alt="{\displaystyle \int _{[0,1]}|(\varphi \circ f)(t)|\,\mathrm {d} t\leq \sum _{n=1}^{\infty }|v_{n}|\cdot n(n+1)\int _{[0,1]}\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }|v_{n}|\cdot n(n+1){\frac {1}{n(n+1)}}=\sum _{n=1}^{\infty }|v_{n}|<\infty }" loading="lazy">.</dd></dl>

Also ist <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"> schwach-integrierbar.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in L^{\infty }([0,1])}">

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<annotation encoding="application/x-tex">{\displaystyle e\in L^{\infty }([0,1])}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32e764fda616df03478dc90a9f1112b13eecb73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.844ex; height:2.843ex;" alt="{\displaystyle e\in L^{\infty }([0,1])}" loading="lazy"> die konstante Funktion mit Wert 1, so ist für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3be1917aa8a9460a761f248e7cb10058b06db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.017ex; height:3.343ex;" alt="{\displaystyle \varphi =(v_{n})_{n}\in \ell ^{1}=c_{0}'}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \langle T_{f}(e),\varphi \rangle =\langle e,S_{f}(\varphi )\rangle =\langle e,\varphi \circ f\rangle =\int _{[0,1]}e(t)(\varphi \circ f)(t)\,\mathrm {d} t=\int _{[0,1]}(\varphi \circ f)(t)\,\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb0a0f2b4797141142309fef7a59798cdc268aa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:78.025ex; height:6.009ex;" alt="{\displaystyle \langle T_{f}(e),\varphi \rangle =\langle e,S_{f}(\varphi )\rangle =\langle e,\varphi \circ f\rangle =\int _{[0,1]}e(t)(\varphi \circ f)(t)\,\mathrm {d} t=\int _{[0,1]}(\varphi \circ f)(t)\,\mathrm {d} t}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle =\int _{[0,1]}\sum _{n=1}^{\infty }v_{n}\cdot n(n+1)\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }v_{n}\cdot n(n+1)\int _{I_{n}}\,\mathrm {d} t=\sum _{n=1}^{\infty }v_{n}=\langle (1,1,1,\ldots ),\varphi \rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48c19084fefb8e91405a695d269fe5de5e02bf8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:86.038ex; height:6.843ex;" alt="{\displaystyle =\int _{[0,1]}\sum _{n=1}^{\infty }v_{n}\cdot n(n+1)\chi _{I_{n}}(t)\,\mathrm {d} t=\sum _{n=1}^{\infty }v_{n}\cdot n(n+1)\int _{I_{n}}\,\mathrm {d} t=\sum _{n=1}^{\infty }v_{n}=\langle (1,1,1,\ldots ),\varphi \rangle }" loading="lazy">.</dd></dl>

Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{[0,1]}f(t)\,\mathrm {d} t=T_{f}(e)=(1,1,1,\ldots )\in \ell ^{\infty }=c_{0}''}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{[0,1]}f(t)\,\mathrm {d} t=T_{f}(e)=(1,1,1,\ldots )\in \ell ^{\infty }=c_{0}''}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ac6c27092ec7de400fd37773c62a35f8e9ef481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.008ex; height:6.009ex;" alt="{\displaystyle \int _{[0,1]}f(t)\,\mathrm {d} t=T_{f}(e)=(1,1,1,\ldots )\in \ell ^{\infty }=c_{0}''}" loading="lazy">

und das ist nicht aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy">. Daher ist <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"> nicht Pettis-integrierbar.<a href="#cite_note-4">[4]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Schwache_Kompaktheit">Schwache Kompaktheit</h3></div>

Ist mit obigen Bezeichnungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> Pettis-integrierbar, so ist der zugehörige Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{f}:L^{\infty }(\Omega ,{\mathcal {A}},\mu )\rightarrow X}">

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<annotation encoding="application/x-tex">{\displaystyle T_{f}:L^{\infty }(\Omega ,{\mathcal {A}},\mu )\rightarrow X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff3b8fefad882cbd32af1824dbb6fb00273cac7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.344ex; height:3.009ex;" alt="{\displaystyle T_{f}:L^{\infty }(\Omega ,{\mathcal {A}},\mu )\rightarrow X}" loading="lazy"> <a href="Schwach-kompakter_Operator" title="Schwach-kompakter Operator">schwach-kompakt</a>.

<div class="mw-heading mw-heading3"><h3 id="Operatoren">Operatoren</h3></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 endlicher, vollständiger Maßraum, <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"> eine Banachraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> Pettis-integrierbar. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:X\rightarrow Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd7c37793df9e7a57df69ff449f84edbac1a6ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.048ex; height:2.176ex;" alt="{\displaystyle A:X\rightarrow Y}" loading="lazy"> ein stetiger, linearer Operator zwischen Banachräumen, so ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\circ f:\Omega \rightarrow Y}">

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<annotation encoding="application/x-tex">{\displaystyle A\circ f:\Omega \rightarrow Y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d82998ba921902c2af94a128588738871dae2a25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.219ex; height:2.509ex;" alt="{\displaystyle A\circ f:\Omega \rightarrow Y}" loading="lazy"> Pettis-integrierbar und es gilt

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6fce2e0f3922346fb632524e98f74b6fdcc7451.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.642ex; height:6.176ex;" alt="{\displaystyle \int _{E}A\circ f\,\mathrm {d} \mu =A\left(\int _{E}f\,\mathrm {d} \mu \right)}" loading="lazy"> für jede messbare Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subset \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fc594930606c44dc02228736fc4bc96d4a8b6f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;" alt="{\displaystyle E\subset \Omega }" loading="lazy">.<a href="#cite_note-5">[5]</a></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Vektorraum_der_Pettis-integrierbaren_Funktionen">Vektorraum der Pettis-integrierbaren Funktionen</h3></div>

Leicht zeigt man, dass Summen und <a href="Skalarmultiplikation" title="Skalarmultiplikation">skalare</a> Vielfache Pettis-integrierbarer Funktionen wieder Pettis-integrierbar sind und dass sich das Integral linear verhält, das heißt

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9f0b4e46ce5f226ec1de9c7e62f28e81dcb423f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.892ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }(\alpha f+g)\,\mathrm {d} \mu =\alpha \int _{\Omega }f\,\mathrm {d} \mu +\int _{\Omega }g\,\mathrm {d} \mu }" loading="lazy"></dd></dl>

für Pettis-integrierbare Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6114e7c2e7ee6dc999cf54bc052d0537e62b8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.638ex; height:2.509ex;" alt="{\displaystyle f,g:\Omega \rightarrow X}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {R} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7988141e89a37e7f4deb883dbd74d9bbd6d11317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {R} }" loading="lazy">.

Die messbaren, Pettis-integrierbaren Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea6cb99c2a8ef5dc7b2b6f3e5b78325677802108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.272ex; height:2.176ex;" alt="{\displaystyle \Omega \rightarrow X}" loading="lazy"> bilden daher einen Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{(1)}(\Omega ,{\mathcal {A}},\mu ,X)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{(1)}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8a429a864a002fe91ffb1251dd754fa9019f5ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.7ex; height:3.176ex;" alt="{\displaystyle P_{(1)}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy">. Die Menge der Funktionen, die <a href="Fast_%C3%BCberall" title="Fast überall">μ-fast-überall</a> den Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/749c1e19d05db6e4efd29c9b6661b99e39e15116.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.983ex; height:2.176ex;" alt="{\displaystyle 0\in X}" loading="lazy"> annehmen, bilden einen Untervektorraum, und den <a href="Faktorraum" title="Faktorraum">Quotientenraum</a> nach diesem Unterraum bezeichnet man mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(\Omega ,{\mathcal {A}},\mu ,X)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{1}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74e160e0955a6eded1ca19643837047d9346a4a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.421ex; height:2.843ex;" alt="{\displaystyle P_{1}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy">. In der maßtheoretisch üblichen Sichtweise ist das der Raum der messbaren, Pettis-integrierbaren Funktionen, wobei μ-fast-überall gleiche Funktionen identifiziert werden.

<div class="mw-heading mw-heading3"><h3 id="Die_1-Norm_für_Pettis-integrierbare_Funktionen">Die 1-Norm für Pettis-integrierbare Funktionen</h3></div>

Ist mit obigen Bezeichnungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \rightarrow X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c88f552b5c49b97290da5a1671ffae8ee127f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:\Omega \rightarrow X}" loading="lazy"> messbar und Pettis-integrierbar, so ist

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<annotation encoding="application/x-tex">{\displaystyle |f|_{1}:=\sup \left\{\int _{\Omega }|\varphi \circ f|\,\mathrm {d} \mu ;\,\varphi \in X',\|\varphi \|\leq 1\right\}=\sup\{T_{f}(g);\,g\in L^{\infty }(\Omega ,{\mathcal {A}},\mu ),\|g\|_{\infty }\leq 1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf941baf35da20873ac64ebf6a87682946764454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.468ex; height:6.176ex;" alt="{\displaystyle |f|_{1}:=\sup \left\{\int _{\Omega }|\varphi \circ f|\,\mathrm {d} \mu ;\,\varphi \in X',\|\varphi \|\leq 1\right\}=\sup\{T_{f}(g);\,g\in L^{\infty }(\Omega ,{\mathcal {A}},\mu ),\|g\|_{\infty }\leq 1\}}" loading="lazy"></dd></dl>

endlich. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7a77d0f6359cf0e4825de67e8a7e9863dcc8410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle |\cdot |_{1}}" loading="lazy"> ist eine <a href="Halbnorm" title="Halbnorm">Halbnorm</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{(1)}(\Omega ,{\mathcal {A}},\mu ,X)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{(1)}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8a429a864a002fe91ffb1251dd754fa9019f5ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.7ex; height:3.176ex;" alt="{\displaystyle P_{(1)}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy"> und eine Norm auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(\Omega ,{\mathcal {A}},\mu ,X)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{1}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74e160e0955a6eded1ca19643837047d9346a4a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.421ex; height:2.843ex;" alt="{\displaystyle P_{1}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy">. Dieser <a href="Normierter_Raum" title="Normierter Raum">normierte Raum</a> ist in der Regel nicht <a href="Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum">vollständig</a>, die Vervollständigung sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/712a32e7b9a55872a79f1e87c3012e198e537a6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.536ex; height:3.509ex;" alt="{\displaystyle {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Injektives_Tensorprodukt">Injektives Tensorprodukt</h3></div>

Es seien wieder <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 endlicher, vollständiger Maßraum und <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"> ein Banachraum. Dann ist

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d7b7322820988d999366a8468f680e0eebb84bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.019ex; height:3.509ex;" alt="{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\times X\rightarrow {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X),\,(f,x)\mapsto f(\cdot )x}" loading="lazy"></dd></dl>

eine <a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilineare Abbildung</a>, und es gilt

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7dcc2d15f8d4f9dc799df906e63ad69f7b112eb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.76ex; height:6.176ex;" alt="{\displaystyle \int _{\Omega }f(\cdot )x\,\mathrm {d} \mu =\left(\int _{\Omega }f\,\mathrm {d} \mu \right)x\in X}" loading="lazy">.</dd></dl>

Diese bilineare Abbildung definiert eine lineare Abbildung auf dem <a href="Tensorprodukt" title="Tensorprodukt">Tensorprodukt</a>

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\otimes X\rightarrow {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/068aa77339a60bc9003f1a083ef933bcaa2186e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.468ex; height:3.509ex;" alt="{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu )\otimes X\rightarrow {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy">.</dd></dl>

Vervollständigt man dieses Tensorprodukt zum <a href="Injektives_Tensorprodukt" title="Injektives Tensorprodukt">injektiven Tensorprodukt</a>, so erhält man einen <a href="Isometrie" title="Isometrie">isometrischen Isomorphismus</a>

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu ){\hat {\otimes }}_{\varepsilon }X\rightarrow {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03070742e091854e39ed812570127f9e312d1043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.434ex; height:3.509ex;" alt="{\displaystyle L^{1}(\Omega ,{\mathcal {A}},\mu ){\hat {\otimes }}_{\varepsilon }X\rightarrow {\overline {P_{1}}}(\Omega ,{\mathcal {A}},\mu ,X)}" loading="lazy">.<a href="#cite_note-6">[6]</a></dd></dl>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer-Verlag 2002, ISBN 1-85233-437-1, Kapitel 3.3: The Dual Space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}(\mu ){\hat {\otimes }}_{\varepsilon }X}">

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<annotation encoding="application/x-tex">{\displaystyle L_{1}(\mu ){\hat {\otimes }}_{\varepsilon }X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83714a56e13588a0120529e66a500b2b93aa464e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.635ex; height:3.009ex;" alt="{\displaystyle L_{1}(\mu ){\hat {\otimes }}_{\varepsilon }X}" loading="lazy"> and the Pettis Integral.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Joseph Diestel: Geometry of Banach Spaces – Selected Topics, Lecture Notes in Mathematics 485, Springer-Verlag (1975), ISBN 3-540-07402-3, Kapitel 6, §1, Theorem 3

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer-Verlag 2002, ISBN 1-85233-437-1, Seite 53

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer-Verlag 2002, ISBN 1-85233-437-1, Seite 52

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer-Verlag 2002, ISBN 1-85233-437-1, Satz 3.7

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces, Springer-Verlag 2002, ISBN 1-85233-437-1, Satz 3.13

</li>

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