ff1588d761d55f48f61a52249e1dcc24

<!DOCTYPE html>

<title>Final-σ-Algebra</title>

</head>

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

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Final-σ-Algebra</b>, auch <b>Bild-σ-Algebra</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> genannt, ist ein spezielles <a href="Mengensystem" title="Mengensystem">Mengensystem</a>, genauer eine <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebra</a>, in der <a href="Ma%C3%9Ftheorie" title="Maßtheorie">Maßtheorie</a>. Die zu einer vorgegebenen <a href="Familie_(Mathematik)" title="Familie (Mathematik)">Familie</a> von <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> gebildete Final-σ-Algebra ist das größte Mengensystem auf der gemeinsamen <a href="Zielmenge" title="Zielmenge">Zielmenge</a> dieser Funktionen, bezüglich der diese sämtlich <a href="Messbare_Funktion" title="Messbare Funktion">messbar</a> sind. Somit bildet das Konzept der Final-σ-Algebra das Pendant zum Konzept der <a href="Initial-%CF%83-Algebra" title="Initial-σ-Algebra">Initial-σ-Algebra</a>, welche die kleinste σ-Algebra auf der <a href="Definitionsmenge" title="Definitionsmenge">Definitionsmenge</a> darstellt, bezüglich der alle Funktionen der vorgegebenen Funktionenfamilie messbar sind. Ein analoges Konzept findet sich in der <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a>; hier sind die <a href="Initialtopologie" title="Initialtopologie">Initialtopologie</a> bzw. die <a href="Finaltopologie" title="Finaltopologie">Finaltopologie</a> die <a href="Gr%C3%B6bere_und_feinere_Topologien" title="Gröbere und feinere Topologien">gröbste bzw. feinste Topologie</a> auf der Definitionsmenge bzw. Zielmenge, bezüglich der alle Funktionen der vorgegebenen Funktionenfamilie <a href="Stetig" class="mw-redirect" title="Stetig">stetig</a> ist.

</p>

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

<p>Für eine beliebige Indexmenge seien <a href="Messraum_(Mathematik)" title="Messraum (Mathematik)">Messräume</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{i},{\mathcal {A}}_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af1eb9539afe228c09ad0ab45395d0f307d21909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.221ex; height:2.843ex;" alt="{\displaystyle (X_{i},{\mathcal {A}}_{i})}" loading="lazy"></span> gegeben sowie Abbildungen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}\colon X_{i}\to X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}\colon X_{i}\to X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84761ab86f372828a9ffc2edddf0f337a1ffe9c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.291ex; height:2.509ex;" alt="{\displaystyle f_{i}\colon X_{i}\to X}" loading="lazy"></span> für eine beliebige Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Dann heißt die <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebra</a>

</p>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f_{i},\,i\in I):=\bigcap _{i\in I}\{A\subset X\,|\,f_{i}^{-1}(A)\in {\mathcal {A}}_{i}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64a7d4fb373bb86b980fdc5117e2d97dd7798577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.506ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}(f_{i},\,i\in I):=\bigcap _{i\in I}\{A\subset X\,|\,f_{i}^{-1}(A)\in {\mathcal {A}}_{i}\}}" loading="lazy"></span></dd></dl>

<p>die <i>Final-σ-Algebra</i> der Abbildungen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"></span> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.

</p>

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

<ul><li>Ist ein weiterer Messraum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y,{\mathcal {A}}^{*})}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f578ce611f1f8da5a850a1035a7f55efa26a096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.589ex; height:2.843ex;" alt="{\displaystyle (Y,{\mathcal {A}}^{*})}" loading="lazy"></span> gegeben und eine Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77d818269c4f3ae277985d810b77037d965e6994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.517ex; height:2.509ex;" alt="{\displaystyle g\colon X\to Y}" loading="lazy"></span>, so ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> genau dann <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f_{i},\,i\in I)}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f_{i},\,i\in I)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ed781467ffd28383c7e9be23ed96ec456dd494c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.911ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(f_{i},\,i\in I)}" loading="lazy"></span> - <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa9d4c5fa60e7b99fec70b03e259a55015d8e7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}^{*}}" loading="lazy"></span>-messbar, wenn die <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Kompositionen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ae3d4ed83fd4bbd7d0bf6f4260ad04cfb8749e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.25ex; height:2.509ex;" alt="{\displaystyle g\circ f_{i}}" loading="lazy"></span> alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a237e3831acdc4229ee8a60fb4d4d764a34567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.654ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{i}}" loading="lazy"></span> - <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa9d4c5fa60e7b99fec70b03e259a55015d8e7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}^{*}}" loading="lazy"></span>-messbar sind.</li></ul>

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

<ul><li><a href="J%C3%BCrgen_Elstrodt" title="Jürgen Elstrodt">Jürgen Elstrodt</a>: <cite style="font-style:italic">Maß- und Integrationstheorie</cite>. 6., korrigierte Auflage. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-540-89728-6">10.1007/978-3-540-89728-6</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Final-%CF%83-Algebra&rft.au=J%C3%BCrgen+Elstrodt&rft.btitle=Ma%C3%9F-+und+Integrationstheorie&rft.date=2009&rft.doi=10.1007%2F978-3-540-89728-6&rft.edition=6.%2C+korrigierte&rft.genre=book&rft.isbn=9783540897279&rft.place=Berlin+Heidelberg&rft.pub=Springer-Verlag" style="display:none"> </span></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Klaus D. Schmidt: <cite style="font-style:italic">Maß und Wahrscheinlichkeit</cite>. 2., durchgesehene Auflage. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>32</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-21026-6">10.1007/978-3-642-21026-6</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Final-%CF%83-Algebra&rft.au=Klaus+D.+Schmidt&rft.btitle=Ma%C3%9F+und+Wahrscheinlichkeit&rft.date=2011&rft.doi=10.1007%2F978-3-642-21026-6&rft.edition=2.%2C+durchgesehene&rft.genre=book&rft.isbn=9783642210259&rft.pages=32&rft.place=Heidelberg+Dordrecht+London+New+York&rft.pub=Springer-Verlag" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2017-09-21" href="https://de.wikipedia.org/wiki/?title=Final-%CF%83-Algebra&oldid=169299859">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>