GermanWikipedia - /page/entry.mu

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<title>Prämaß</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Prämaß</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Ein Prämaß ist eine spezielle <a href="Mengenfunktion" title="Mengenfunktion">Mengenfunktion</a> in der <a href="Ma%C3%9Ftheorie" title="Maßtheorie">Maßtheorie</a>, die verwendet wird, um den intuitiven Volumenbegriff mathematisch zu präzisieren. Im Gegensatz zu einem <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Maß</a> muss der Definitionsbereich eines Prämaßes keine <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebra</a> sein.

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

Eine Mengenfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu :{\mathcal {C}}\to [0,\infty ]}">

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<annotation encoding="application/x-tex">{\displaystyle \mu :{\mathcal {C}}\to [0,\infty ]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e6ea20cff9642fed34b67c6f1e7f1d976a3751b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.006ex; height:2.843ex;" alt="{\displaystyle \mu :{\mathcal {C}}\to [0,\infty ]}" loading="lazy"> von dem Mengensystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> heißt Prämaß, wenn sie die folgenden Eigenschaften erfüllt:

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<annotation encoding="application/x-tex">{\displaystyle \mu (\emptyset )=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62729938e4e5129afccae3e4795f9f10437f408c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.634ex; height:2.843ex;" alt="{\displaystyle \mu (\emptyset )=0}" loading="lazy"></li>

<li>Sie ist <a href="%CE%A3-Additivit%C3%A4t" title="Σ-Additivität">σ-additiv</a>, das heißt für jede Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{i})_{i\in \mathbb {N} }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5dff72e778db69131292b8d10885f5afb1b3170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:2.843ex;" alt="{\displaystyle (A_{i})_{i\in \mathbb {N} }}" loading="lazy"> <a href="Abz%C3%A4hlbarkeit" class="mw-redirect" title="Abzählbarkeit">abzählbar</a> vieler <a href="Paarweise_disjunkt" class="mw-redirect" title="Paarweise disjunkt">paarweise disjunkter</a> Mengen aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {C}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {C}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fc81af6e880a32375d86cd94a3a97d849171f90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.846ex; height:3.176ex;" alt="{\displaystyle \textstyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {C}}}" loading="lazy"> gilt:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mu (A_{i}).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b21b1980636241941237eefbd29ff8b85c42463.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.309ex; height:7.509ex;" alt="{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mu (A_{i}).}" loading="lazy"></dd></dl>

Alternativ lässt sich ein Prämaß auch als ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-additiver <a href="Inhalt_(Ma%C3%9Ftheorie)" title="Inhalt (Maßtheorie)">Inhalt</a> definieren. Als Mengensystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> wählt man meist einen <a href="Halbring_(Mengensystem)" title="Halbring (Mengensystem)">Halbring</a> oder einen <a href="Ring_(Mengensystem)" title="Ring (Mengensystem)">Ring</a>.

Ein Prämaß heißt endlich, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A)<\infty }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a97dfa93ce05b5c7fbac7bec0f21dfd8cbde288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.376ex; height:2.843ex;" alt="{\displaystyle \mu (A)<\infty }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/104c463254e5aee10373097f39368b0d53757f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.823ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {C}}}" loading="lazy"> gilt. Ein Prämaß heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5dff72e778db69131292b8d10885f5afb1b3170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:2.843ex;" alt="{\displaystyle (A_{i})_{i\in \mathbb {N} }}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A_{i})<\infty }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2c0f78cbc19b58c7ee017633fd54a98ba2d8aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.176ex; height:2.843ex;" alt="{\displaystyle \mu (A_{i})<\infty }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {N} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e64c8c5906eb3eb9d7a8b1ed1e31de4e5fc6c632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.321ex; height:2.176ex;" alt="{\displaystyle i\in \mathbb {N} }" loading="lazy"> gilt.

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

Je nachdem, auf welchem <a href="Mengensystem" title="Mengensystem">Mengensystem</a> ein Prämaß definiert ist, ändern sich die Eigenschaften. Zusätzlich zu allen hier genannten Eigenschaften gelten auch alle Eigenschaften von Inhalten auf dem entsprechenden Mengensystem.

<div class="mw-heading mw-heading3"><h3 id="Im_Halbring">Im Halbring</h3></div>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5dc90bfe670940db7669cead3e1e98c07e58f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}={\mathcal {H}}}" loading="lazy"> ein <a href="Halbring_(Mengensystem)" title="Halbring (Mengensystem)">Halbring</a> ist, dann kann man zu jedem Prämaß <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"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"> ein eindeutiges Prämaß <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_/eb734a37dd21ce173a46342d1cc64c92/53b9ddd6bbc6f1ef9ca40b1c2e6e2c1c8d141aed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.086ex; height:3.009ex;" alt="{\displaystyle \mu '}" loading="lazy"> auf dem von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"> erzeugten <a href="Ring_(Mengensystem)" title="Ring (Mengensystem)">Ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> konstruieren. Siehe hierzu auch den Abschnitt über Fortsetzungen.

Falls <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"> ein Inhalt auf einem <a href="Ring_(Mengensystem)" title="Ring (Mengensystem)">Ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}={\mathcal {R}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77f3fea155e5220dbc3e1702b467551c8f2246ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.308ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}={\mathcal {R}}}" loading="lazy"> ist, dann gelten bei den folgende Eigenschaften die Implikationen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\iff 2\iff 3\Longrightarrow 4\iff 5}">

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<annotation encoding="application/x-tex">{\displaystyle 1\iff 2\iff 3\Longrightarrow 4\iff 5}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c819fab1a5ecb9b2ae2743a2f73edfdca584032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:31.6ex; height:2.176ex;" alt="{\displaystyle 1\iff 2\iff 3\Longrightarrow 4\iff 5}" loading="lazy">

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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"> ist ein Prämaß.</li>

<li><a href="%CE%A3-Subadditivit%C3%A4t" title="Σ-Subadditivität">σ-subadditiv</a> (Sigma-subadditiv), es gilt folglich: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \left(\bigcup _{j\in \mathbb {N} }A_{j}\right)\leq \sum _{j\in \mathbb {N} }\mu (A_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle \mu \left(\bigcup _{j\in \mathbb {N} }A_{j}\right)\leq \sum _{j\in \mathbb {N} }\mu (A_{j})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33dfc3c66372ba99e0f61435084bc54cb44ee5ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.175ex; height:7.676ex;" alt="{\displaystyle \mu \left(\bigcup _{j\in \mathbb {N} }A_{j}\right)\leq \sum _{j\in \mathbb {N} }\mu (A_{j})}" loading="lazy"> für jede <a href="Folge_(Mathematik)" title="Folge (Mathematik)">Folge</a> von Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{j})_{j\in \mathbb {N} }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aaa50d310f3360bea52dd400158ebad73f37a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.655ex; height:3.009ex;" alt="{\displaystyle (A_{j})_{j\in \mathbb {N} }}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}">

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<annotation encoding="application/x-tex">{\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5871dd74e6b65e95dfb71aa40d0093975c1cab22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:10.099ex; height:5.843ex;" alt="{\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}" loading="lazy"></li>

<li><a href="Stetigkeit_von_unten" class="mw-redirect" title="Stetigkeit von unten">Stetigkeit von unten</a>: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{i})_{i\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5dff72e778db69131292b8d10885f5afb1b3170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:2.843ex;" alt="{\displaystyle (A_{i})_{i\in \mathbb {N} }}" loading="lazy"> eine gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13fe498478b50d9853e11396a3d2d71317c440cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.554ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {R}}}" loading="lazy"> aufsteigende Folge von Mengen aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A_{i})\uparrow \mu (A)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1331c3bc4aeb59069ac8882c2920063f45a61b5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.161ex; height:2.843ex;" alt="{\displaystyle \mu (A_{i})\uparrow \mu (A)}" loading="lazy">.</li>

<li><a href="Stetigkeit_von_oben" class="mw-redirect" title="Stetigkeit von oben">Stetigkeit von oben</a>: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{i})_{i\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5dff72e778db69131292b8d10885f5afb1b3170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:2.843ex;" alt="{\displaystyle (A_{i})_{i\in \mathbb {N} }}" loading="lazy"> eine gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13fe498478b50d9853e11396a3d2d71317c440cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.554ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {R}}}" loading="lazy"> absteigende Folge von Mengen aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A_{1})<\infty }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (A_{1})<\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48132603ddd6eadad9676934cd94cb9fab54c05b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.431ex; height:2.843ex;" alt="{\displaystyle \mu (A_{1})<\infty }" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A_{i})\downarrow \mu (A)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed7527f69405a5544602d74297d2ba74cc017b69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.161ex; height:2.843ex;" alt="{\displaystyle \mu (A_{i})\downarrow \mu (A)}" loading="lazy">.</li>

<li>Stetigkeit gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6af50205f42bb2ec3c666b7b847d2c7f96e464c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }" loading="lazy">: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{i})_{i\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5dff72e778db69131292b8d10885f5afb1b3170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:2.843ex;" alt="{\displaystyle (A_{i})_{i\in \mathbb {N} }}" loading="lazy"> eine gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6af50205f42bb2ec3c666b7b847d2c7f96e464c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }" loading="lazy"> absteigende Folge von Mengen aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A_{1})<\infty }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (A_{1})<\infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48132603ddd6eadad9676934cd94cb9fab54c05b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.431ex; height:2.843ex;" alt="{\displaystyle \mu (A_{1})<\infty }" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A_{i})\downarrow 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (A_{i})\downarrow 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd66193489e6c35a9cdcc56ec21360946009d027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.369ex; height:2.843ex;" alt="{\displaystyle \mu (A_{i})\downarrow 0}" loading="lazy">.</li></ol>

Diese Eigenschaften werden auch oft als Charakterisierung genutzt. Ist das Prämaß endlich, so gilt bei allen Eigenschaften Äquivalenz.

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

<div class="mw-heading mw-heading3"><h3 id="Von_Halbringen_auf_Ringe">Von Halbringen auf Ringe</h3></div>

Man kann zu jedem Prämaß <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"> auf dem Halbring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"> ein Prämaß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu '}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53b9ddd6bbc6f1ef9ca40b1c2e6e2c1c8d141aed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.086ex; height:3.009ex;" alt="{\displaystyle \mu '}" loading="lazy"> auf dem von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"> erzeugten <a href="Ring_(Mengensystem)" title="Ring (Mengensystem)"> Ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> konstruieren. Aufgrund der Eigenschaften eines Halbringes gibt es für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13fe498478b50d9853e11396a3d2d71317c440cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.554ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {R}}}" loading="lazy"> paarweise disjunkte Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},A_{2},\dotsc ,A_{m}\in {\mathcal {H}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},\dotsc ,A_{m}\in {\mathcal {H}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5743ad0aec55a4285e21cfab7b151143b23e031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.029ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},\dotsc ,A_{m}\in {\mathcal {H}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A=\bigcup _{j=1}^{m}A_{j}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle A=\bigcup _{j=1}^{m}A_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a68e52b59ba6b690829b0172d229c45139aefdd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.828ex; height:3.509ex;" alt="{\displaystyle \textstyle A=\bigcup _{j=1}^{m}A_{j}}" loading="lazy">. Indem man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu '}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53b9ddd6bbc6f1ef9ca40b1c2e6e2c1c8d141aed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.086ex; height:3.009ex;" alt="{\displaystyle \mu '}" loading="lazy"> durch

<msup>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu '(A):=\sum _{j=1}^{m}\mu (A_{j})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c22d679c8eaa57c1bb2e96f35cefa5869323d756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.99ex; height:7.176ex;" alt="{\displaystyle \mu '(A):=\sum _{j=1}^{m}\mu (A_{j})}" loading="lazy"></dd></dl>

definiert, erhält man eine eindeutig bestimmte Fortsetzung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu '}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53b9ddd6bbc6f1ef9ca40b1c2e6e2c1c8d141aed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.086ex; height:3.009ex;" alt="{\displaystyle \mu '}" loading="lazy">. Die Fortsetzung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu '}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53b9ddd6bbc6f1ef9ca40b1c2e6e2c1c8d141aed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.086ex; height:3.009ex;" alt="{\displaystyle \mu '}" loading="lazy"> ist genau dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

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

<div class="mw-heading mw-heading3"><h3 id="Zu_einem_Maß">Zu einem Maß</h3></div>

Nach dem <a href="Ma%C3%9Ferweiterungssatz_von_Carath%C3%A9odory" title="Maßerweiterungssatz von Carathéodory">Maßerweiterungssatz von Carathéodory</a> kann ein Prämaß auf einem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> zu einem Maß auf der vom Ring erzeugten <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebra</a> fortgesetzt werden. Dazu wird aus dem Prämaß zuerst ein <a href="%C3%84u%C3%9Feres_Ma%C3%9F" title="Äußeres Maß">äußeres Maß</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> konstruiert. Diejenigen Mengen, die <a href="Messbarkeit_nach_Carath%C3%A9odory" class="mw-redirect" title="Messbarkeit nach Carathéodory">bezüglich dieses äußeren Maßes messbar</a> sind bilden eine σ-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}_{\nu }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/431179fc9fa416cb64877ed487048def4f4cc224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.958ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{\nu }}" loading="lazy">. Die Einschränkung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}">

<msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f11abffb49355cb307eb4f9ba9ef981b26d6f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.294ex; height:3.176ex;" alt="{\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}" loading="lazy"> des äußeren Maßes auf diese σ-Algebra ist dann ein Maß, dass auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> mit dem Prämaß übereinstimmt. Des Weiteren enthält <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}_{\nu }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/431179fc9fa416cb64877ed487048def4f4cc224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.958ex; height:2.676ex;" alt="{\displaystyle {\mathcal {A}}_{\nu }}" loading="lazy"> den Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"> und damit auch die von dem Ring erzeugte σ-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ({\mathcal {R}})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8269a805e04c87a030329fda4d20105741a1b60b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.11ex; height:2.843ex;" alt="{\displaystyle \sigma ({\mathcal {R}})}" loading="lazy">.

Außerdem ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {A}}_{\nu },\nu |_{{\mathcal {A}}_{\nu }})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</mrow>

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</mrow>

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e322f0a99cfe56034a72828c3dcd555ac51f771.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.808ex; height:3.176ex;" alt="{\displaystyle (\Omega ,{\mathcal {A}}_{\nu },\nu |_{{\mathcal {A}}_{\nu }})}" loading="lazy"> ein <a href="Vollst%C3%A4ndiger_Ma%C3%9Fraum" class="mw-redirect" title="Vollständiger Maßraum">Vollständiger Maßraum</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}">

<msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f11abffb49355cb307eb4f9ba9ef981b26d6f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.294ex; height:3.176ex;" alt="{\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}" loading="lazy"> ist die <a href="Vollst%C3%A4ndiges_Ma%C3%9F" title="Vollständiges Maß">Vervollständigung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu |_{\sigma ({\mathcal {R}})}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu |_{\sigma ({\mathcal {R}})}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dafc55e83f42de8e07c627bfe1769e485eac8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.724ex; height:3.343ex;" alt="{\displaystyle \nu |_{\sigma ({\mathcal {R}})}}" loading="lazy">.

Ist das Prämaß σ-endlich, so folgt mit dem <a href="Ma%C3%9F_(Mathematik)#Eindeutigkeitssatz" title="Maß (Mathematik)">Eindeutigkeitssatz für Maße</a> die Eindeutigkeit der Fortsetzung.

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

<div class="mw-heading mw-heading3"><h3 id="Lebesguesches_Prämaß">Lebesguesches Prämaß</h3></div>

Das wichtigste Prämaß ist das sogenannte <a href="Lebesguesches_Pr%C3%A4ma%C3%9F" title="Lebesguesches Prämaß">Lebesgue’sche Prämaß</a>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \lambda ([a,b))=b-a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/faec7079369c82101b716844432c2a1108d243c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.144ex; height:2.843ex;" alt="{\displaystyle \lambda ([a,b))=b-a}" loading="lazy">.</dd></dl>

auf dem Halbring der halboffenen Intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle [a,b)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b0ffcbb2f26f5db44992d0c8cc2a3717b5e815e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.813ex; height:2.843ex;" alt="{\displaystyle [a,b)}" loading="lazy"> auf den reellen Zahlen. Es lässt sich auch auf höhere Dimensionen verallgemeinern. Aus ihm wird das <a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a> und anschließend das <a href="Lebesgue-Integral" title="Lebesgue-Integral">Lebesgue-Integral</a> konstruiert.

<div class="mw-heading mw-heading3"><h3 id="Lebesgue-Stieltjessches_Prämaß">Lebesgue-Stieltjessches Prämaß</h3></div>

Ein weiteres wichtiges Prämaß ist das <a href="Lebesgue-Stieltjessches_Pr%C3%A4ma%C3%9F" class="mw-redirect" title="Lebesgue-Stieltjessches Prämaß">Lebesgue-Stieltjesches Prämaß</a>, aus dem sich das <a href="Lebesgue-Stieltjes-Ma%C3%9F" title="Lebesgue-Stieltjes-Maß">Lebesgue-Stieltjes-Maß</a> und das <a href="Lebesgue-Stieltjes-Integral" class="mw-redirect" title="Lebesgue-Stieltjes-Integral">Lebesgue-Stieltjes-Integral</a> ableitet:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{F}([a,b))=F(b)-F(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42c5c671f960f37e958e4a02612bb504e6ba4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.753ex; height:2.843ex;" alt="{\displaystyle \mu _{F}([a,b))=F(b)-F(a)}" loading="lazy">,</dd></dl>

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

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> eine wachsende rechtsseitig stetige reellwertige Funktion ist. 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_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> nicht rechtsseitig stetig, so handelt es sich um den <a href="Stieltjes%E2%80%99scher_Inhalt" class="mw-redirect" title="Stieltjes’scher Inhalt">Stieltjes’schen Inhalt</a>. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44e1cf18eb890214752fa45602f92cc8e8b85f89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.308ex; height:2.843ex;" alt="{\displaystyle F(x)=x}" loading="lazy"> stimmt es mit dem Lebesgueschen Prämaß überein. Jedes endliche Prämaß auf den Reellen Zahlen kann als Lebesgue-Stieltjessches Prämaß dargestellt werden mit einer passenden 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_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">.

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

<ul><li>Jürgen Elstrodt: Maß- und Integrationstheorie. 6. Auflage. Springer, Berlin/Heidelberg/New York 2009, ISBN 978-3-540-89727-9.</li>

<li>Achim Klenke: Wahrscheinlichkeitstheorie. 2. Auflage. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8</li></ul></div><div><div class="zim-footer">

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