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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Spur eines Mengensystems</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Spur eines Mengensystems</b> ist ein Begriff aus der <a href="Mathematik" title="Mathematik">Mathematik</a> und wird insbesondere häufig in der <a href="Ma%C3%9Ftheorie" title="Maßtheorie">Maßtheorie</a> und der <a href="Stochastik" title="Stochastik">Stochastik</a> verwendet. Er beschreibt die Reduktion von <a href="Mengensystem" title="Mengensystem">Mengensystemen</a> auf eine kleinere Grundmenge und ist eng verwandt mit dem Begriff der <a href="Spurtopologie" class="mw-redirect" title="Spurtopologie">Spurtopologie</a>.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Gegeben sei ein beliebiges Mengensystem <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 {M}}\subset {\mathcal {P}}(\Omega )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">M</mi>
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<mo>⊂<!-- ⊂ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
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<mo stretchy="false">(</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}\subset {\mathcal {P}}(\Omega )}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/020ba37edd5b5d7b58fc8adf475574c620d909b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.08ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}\subset {\mathcal {P}}(\Omega )}" loading="lazy"></span> auf der Grundmenge <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 \Omega }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> und eine 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 E\subset \Omega }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>⊂<!-- ⊂ --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
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<annotation encoding="application/x-tex">{\displaystyle E\subset \Omega }</annotation>
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</math></span><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"></span>. Dann heißt
</p>
<dl><dd><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 {M}}|_{E}:=\{M\cap E\,|\,M\in {\mathcal {M}}\}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">M</mi>
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>E</mi>
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<mo>:=</mo>
<mo fence="false" stretchy="false">{</mo>
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<mo>∩<!-- ∩ --></mo>
<mi>E</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">M</mi>
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<mo fence="false" stretchy="false">}</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}|_{E}:=\{M\cap E\,|\,M\in {\mathcal {M}}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d75e9edea911e8f56c242b36678497f96f01ed54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.29ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}|_{E}:=\{M\cap E\,|\,M\in {\mathcal {M}}\}}" loading="lazy"></span></dd></dl>
<p>die Spur oder Einschränkung von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">M</mi>
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</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" 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 E}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
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</math></span><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"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Bemerkung">Bemerkung</h2></div>
<p>Im Allgemeinen ist die Spur eines Mengensystems nicht mehr vom selben Typ wie das ursprüngliche Mengensystem. Beispiel hierfür sind <a href="Dynkin-System" title="Dynkin-System">Dynkin-Systeme</a>. Klassen von Mengensystemen, deren Spur wieder von derselben Klasse ist, sind <a href="Halbring_(Mengensystem)" title="Halbring (Mengensystem)">Halbringe</a>, <a href="Mengenring" class="mw-redirect" title="Mengenring">Mengenringe</a>, <a href="Algebra_(Mengensystem)" title="Algebra (Mengensystem)">Mengenalgebren</a> und <a href="%CE%A3-Ring" title="Σ-Ring">σ-Ringe</a> sowie <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebren</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Sei <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 \Omega =\{1,2,3\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega =\{1,2,3\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd8806c6bec78bb9ee6b164d3632a0d25bbc4a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.657ex; height:2.843ex;" alt="{\displaystyle \Omega =\{1,2,3\}}" loading="lazy"></span>, eine dazugehörige σ-Algebra <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}}=\{\emptyset ,\{1\},\{2,3\},\Omega \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">A</mi>
</mrow>
</mrow>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}=\{\emptyset ,\{1\},\{2,3\},\Omega \}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/791bf31608166473a63ecd200ed392a4cae5ae59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.44ex; height:2.843ex;" alt="{\displaystyle {\mathcal {A}}=\{\emptyset ,\{1\},\{2,3\},\Omega \}}" loading="lazy"></span> und <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 E=\{1,2\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E=\{1,2\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54ba40d791e6b1ba07c6c4b490453e836258dc8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.558ex; height:2.843ex;" alt="{\displaystyle E=\{1,2\}}" 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 {\mathcal {A}}|_{E}=\{\emptyset ,\{1\},\{2\},E\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">A</mi>
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<mi>E</mi>
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</msub>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">∅<!-- ∅ --></mi>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>2</mn>
<mo fence="false" stretchy="false">}</mo>
<mo>,</mo>
<mi>E</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}|_{E}=\{\emptyset ,\{1\},\{2\},E\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/febef8ce855855bf8ceb9672ae550b7dca59adda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.476ex; height:2.843ex;" alt="{\displaystyle {\mathcal {A}}|_{E}=\{\emptyset ,\{1\},\{2\},E\}}" loading="lazy"></span> die <a href="Spur-%CF%83-Algebra" class="mw-redirect" title="Spur-σ-Algebra">Spur-σ-Algebra</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">A</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation>
</semantics>
</math></span><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"></span> über <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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
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</math></span><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"></span>.
</p>
<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>: <i>Maß- und Integrationstheorie.</i> 4., korrigierte Auflage. Springer, Berlin u. a. 2005, ISBN 3-540-21390-2</li>
<li>Achim Klenke: <i>Wahrscheinlichkeitstheorie.</i> 2. Auflage. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="cite">Achim Klenke: <a rel="nofollow" class="external text" href="http://www.spektrum.de/sixcms/media.php/370/Leseprobe.140140.pdf"><i>Wahrscheinlichkeitstheorie.</i></a> (PDF) <span style="white-space:nowrap;">S. 10</span>,<span class="Abrufdatum"> abgerufen am 4. Januar 2015</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ASpur+eines+Mengensystems&rft.title=Wahrscheinlichkeitstheorie&rft.description=Wahrscheinlichkeitstheorie&rft.identifier=http%3A%2F%2Fwww.spektrum.de%2Fsixcms%2Fmedia.php%2F370%2FLeseprobe.140140.pdf&rft.creator=Achim+Klenke"> </span></span>
</li>
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