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<title>Halbring (Mengensystem)</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Halbring (Mengensystem)</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 (Mengen-)Halbring, auch (Mengen-)Semiring genannt, ist ein spezielles <a href="Mengensystem" title="Mengensystem">Mengensystem</a> in der <a href="Ma%C3%9Ftheorie" title="Maßtheorie">Maßtheorie</a>, einem Teilgebiet der Mathematik, welches die Grundlage für die moderne <a href="Integralrechnung" title="Integralrechnung">Integrationstheorie</a> und <a href="Stochastik" title="Stochastik">Stochastik</a> bildet.

Aufgrund ihrer guten Handhabbarkeit werden Halbringe beispielsweise als Definitionsbereiche von <a href="Inhalt_(Ma%C3%9Ftheorie)" title="Inhalt (Maßtheorie)">Inhalten</a> verwendet, die dann schrittweise zu <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Maßen</a> erweitert werden. Ebenso sind sie beliebte <a href="Erzeuger_einer_%CF%83-Algebra" class="mw-redirect" title="Erzeuger einer σ-Algebra">Erzeuger von σ-Algebren</a>, insbesondere der <a href="Borelsche_%CF%83-Algebra" title="Borelsche σ-Algebra">Borelschen σ-Algebra</a>, da nach dem <a href="Ma%C3%9Feindeutigkeitssatz" title="Maßeindeutigkeitssatz">Maßeindeutigkeitssatz</a> ein Maß durch seine Werte auf einem Halbring bereits auf der erzeugten σ-Algebra eindeutig festgelegt ist, sofern das Maß σ-endlich über dem Halbring ist.

Die Definition wurde von <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> als Verallgemeinerung eines <a href="Ring_(Mengensystem)" title="Ring (Mengensystem)">Mengenrings</a> eingeführt.<a href="#cite_note-1">[1]</a> Der hier verwendete Begriff des Halbrings unterscheidet sich grundlegend von dem eines <a href="Halbring_(Algebraische_Struktur)" class="mw-redirect" title="Halbring (Algebraische Struktur)">Halbrings im Sinne der Algebra</a>, also einer speziellen <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraischen Struktur</a>. Beide stehen nicht in engem Zusammenhang!

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

Sei <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"> eine beliebige Menge. Ein <a href="Mengensystem" title="Mengensystem">Mengensystem</a> <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"> von Teilmengen 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"> heißt ein Mengenhalbring oder Halbring über <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">, wenn folgende drei Eigenschaften erfüllt sind:<a href="#cite_note-2">[2]</a>

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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"> enthält die leere Menge: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset \in {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06d73d359ffb710f590e1ec9a60bda09b382d9a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.967ex; height:2.509ex;" alt="{\displaystyle \emptyset \in {\mathcal {H}}}" loading="lazy"></li>

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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"> ist <a href="Durchschnittsstabil" class="mw-redirect" title="Durchschnittsstabil">durchschnittsstabil</a>, das heißt, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d3b84391bd2f3961540b63b1f7e9a56b567d5b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.548ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {H}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28cff1710ca88fb661aa92b590749f14b6889dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.568ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {H}}}" loading="lazy">, so ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B\in {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e47562582e828f963b09e14eb59b5fa8da7c1e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.894ex; height:2.176ex;" alt="{\displaystyle A\cap B\in {\mathcal {H}}}" loading="lazy"></li>

<li>Die <a href="Mengenlehre#Differenz_und_Komplement" title="Mengenlehre">Differenz</a> zweier Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"> aus <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"> lässt sich als endliche Vereinigung von paarweise disjunkten Mengen aus <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"> darstellen. Es existieren also immer paarweise disjunkte Mengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},C_{2},\dotsc ,C_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1},C_{2},\dotsc ,C_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cee76a6e56cc95824a843f84f92b0bf157d0078e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.525ex; height:2.509ex;" alt="{\displaystyle C_{1},C_{2},\dotsc ,C_{n}}" loading="lazy"> aus <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">, sodass</li></ol>

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<annotation encoding="application/x-tex">{\displaystyle A\setminus B=\bigcup _{i=1}^{n}C_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d52a73d10da0dbe06d4681270ed8c783e9626b6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.317ex; height:6.843ex;" alt="{\displaystyle A\setminus B=\bigcup _{i=1}^{n}C_{i}}" loading="lazy">.</dd></dl></dd></dl></dd></dl>

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

Über jeder beliebigen Menge <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"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{1}=\{\emptyset \}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{1}=\{\emptyset \}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dea1c4e62a66e0ae272aa3e17948ade18f8a5e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.604ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{1}=\{\emptyset \}}" loading="lazy"> der kleinste und die <a href="Potenzmenge" title="Potenzmenge">Potenzmenge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{2}={\mathcal {P}}(\Omega )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{2}={\mathcal {P}}(\Omega )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02ee5cd17dd53333aac8b9a5fbe7045e9c0bca4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.307ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{2}={\mathcal {P}}(\Omega )}" loading="lazy"> der größte mögliche Mengenhalbring. Beide enthalten trivialerweise die leere Menge. Der Halbring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b3de4facc7bd3250b910c4363111ae952f7446f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{1}}" loading="lazy"> ist schnittstabil, da die leere Menge mit sich selbst geschnitten wieder die leere Menge ist. Dasselbe gilt für die Differenz der leeren Menge mit sich selbst. Die Aussagen für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a40a9c1d9df2f248ea15869968cef102c511684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{2}}" loading="lazy"> folgen aus der Tatsache, dass die Potenzmenge alle Teilmengen enthält und daher stabil gegenüber allen Mengenoperationen ist.

Ein in der Anwendung wichtiger Halbring über den reellen Zahlen <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"> ist das Mengensystem der endlichen, rechts halboffenen <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervalle</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}:=\{[a,b)\mid a,b\in \mathbb {R} ,a\leq b\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a40c1ec5d6491ce8ba335ff4e35e7064c6928ada.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.069ex; width:28.521ex; height:2.843ex;" alt="{\displaystyle {\mathcal {I}}:=\{[a,b)\mid a,b\in \mathbb {R} ,a\leq b\}}" loading="lazy">.</dd></dl>

Halbringe dieser Art werden häufig als <a href="Erzeuger_einer_%CF%83-Algebra" class="mw-redirect" title="Erzeuger einer σ-Algebra">Erzeuger</a> für die <a href="Borelsche_%CF%83-Algebra" title="Borelsche σ-Algebra">Borelsche σ-Algebra</a> auf <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"> gewählt, teils mit leichten Abwandlungen (links offene, rechts geschlossene Intervalle, nur rationale Grenzen etc.).

Halbringe dieser Art lassen sich auch auf dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> formulieren, wo sie ebenfalls als Erzeuger für die Borelsche σ-Algebra auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> dienen. Setzt man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{1},a_{2},\dotsc ,a_{n})\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a=(a_{1},a_{2},\dotsc ,a_{n})\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cbe934c0a87e70159c0823a161237555fce49b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.103ex; height:2.843ex;" alt="{\displaystyle a=(a_{1},a_{2},\dotsc ,a_{n})\in \mathbb {R} ^{n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=(b_{1},b_{2},\dotsc ,b_{n})\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b=(b_{1},b_{2},\dotsc ,b_{n})\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86ddc37b887ef059d3b6ff21e51c3c845e04571a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.174ex; height:2.843ex;" alt="{\displaystyle b=(b_{1},b_{2},\dotsc ,b_{n})\in \mathbb {R} ^{n}}" loading="lazy"> als Intervalle

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<annotation encoding="application/x-tex">{\displaystyle [a,b)=\{x\in \mathbb {R} ^{n}\;\mid \;a_{1}\leq x_{1}<b_{1},\;a_{2}\leq x_{2}<b_{2},\;\dotsc ,a_{n}\leq x_{n}<b_{n}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df28b3e6cbe4552f458bebbd21d850c80e401814.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.276ex; height:2.843ex;" alt="{\displaystyle [a,b)=\{x\in \mathbb {R} ^{n}\;\mid \;a_{1}\leq x_{1}<b_{1},\;a_{2}\leq x_{2}<b_{2},\;\dotsc ,a_{n}\leq x_{n}<b_{n}\}}" loading="lazy"></dd></dl>

und definiert

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41558abc50886fdf38817495b243958d7b3dd39b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}" loading="lazy"> genau dann, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\leq b_{i}}">

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</msub>

<msub>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa0aab1427302e206a6e4574a72861a71641fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.925ex; height:2.509ex;" alt="{\displaystyle a_{i}\leq b_{i}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dotsc ,n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7f2132430a61b900cf2c4380774394ca9f09c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,n}" loading="lazy">,</dd></dl>

so ist

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}^{n}:=\{[a,b)\mid a,b\in \mathbb {R} ^{n},a\leq b\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afb94b2c5dbd8742814fd18efb5080b9422955c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.069ex; width:31.025ex; height:2.843ex;" alt="{\displaystyle {\mathcal {I}}^{n}:=\{[a,b)\mid a,b\in \mathbb {R} ^{n},a\leq b\}}" loading="lazy"></dd></dl>

ein Halbring, der aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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

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</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">-dimensionalen endlichen, rechts halboffenen Intervallen (<a href="Quader" title="Quader">Quadern</a>) besteht. Ein Spezialfall hiervon sind die <a href="Dyadische_Elementarzellen" title="Dyadische Elementarzellen">dyadischen Elementarzellen</a>. Hier liegen die Eckpunkte der Intervalle alle auf einem Gitter.

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

Aus der Durchschnittsstabilität folgt <a href="Vollst%C3%A4ndige_Induktion" title="Vollständige Induktion">induktiv</a>, dass auch jeder nichtleere, endliche Durchschnitt von Elementen des Mengenhalbrings <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"> in ihm enthalten ist, d. h., für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> gilt:

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<annotation encoding="application/x-tex">{\displaystyle A_{1},\dotsc ,A_{n}\in {\mathcal {H}}\Rightarrow A_{1}\cap \dotsb \cap A_{n}\in {\mathcal {H}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebb193833051c45933f7a5325637980909696cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.454ex; height:2.509ex;" alt="{\displaystyle A_{1},\dotsc ,A_{n}\in {\mathcal {H}}\Rightarrow A_{1}\cap \dotsb \cap A_{n}\in {\mathcal {H}}.}" loading="lazy"></dd></dl>

Mengenhalbringe treten insbesondere als <a href="Erzeugendensystem" title="Erzeugendensystem">Erzeugendensysteme</a> von <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebren</a> auf. Aufgrund der <a href="Durchschnittsstabil" class="mw-redirect" title="Durchschnittsstabil">Durchschnittsstabilität</a> der Halbringe folgt dabei nach dem <a href="Dynkinscher_%CF%80-%CE%BB-Satz" title="Dynkinscher π-λ-Satz">Dynkinschen π-λ-Satz</a>, dass die von einem Halbring <a href="Erzeuger_einer_%CF%83-Algebra" class="mw-redirect" title="Erzeuger einer σ-Algebra">erzeugte σ-Algebra</a> gleich dem <a href="Erzeugtes_Dynkin-System" class="mw-redirect" title="Erzeugtes Dynkin-System">erzeugten Dynkin-System</a> ist, es gilt also

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6909a9beca1f3cf81152ee1510db4adc8a65293a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.023ex; height:2.843ex;" alt="{\displaystyle \sigma ({\mathcal {H}})=\delta ({\mathcal {H}})}" loading="lazy">.</dd></dl>

Ebenso sind daher nach dem <a href="Ma%C3%9Feindeutigkeitssatz" title="Maßeindeutigkeitssatz">Maßeindeutigkeitssatz</a> Maße bereits durch die Angabe ihrer Werte auf dem Halbring eindeutig bestimmt.

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

<div class="mw-heading mw-heading3"><h3 id="Schnitte_von_Halbringen">Schnitte von Halbringen</h3></div>

Im Gegensatz zu den meisten Mengensystemen der Maßtheorie ist der Schnitt von Halbringen, also das Mengensystem

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{1}\cap {\mathcal {H}}_{2}=\{A\subset \Omega \;|\;A\in {\mathcal {H}}_{1}{\text{ und }}A\in {\mathcal {H}}_{2}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42b135f02956d48c6eea1b07b7cc3e5be3af7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.741ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{1}\cap {\mathcal {H}}_{2}=\{A\subset \Omega \;|\;A\in {\mathcal {H}}_{1}{\text{ und }}A\in {\mathcal {H}}_{2}\}}" loading="lazy"></dd></dl>

im Allgemeinen kein Halbring. Gegenbeispiel sind die Halbringe

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{1}=\{\emptyset ,\{1\},\{4\},\{2,3\},\{1,2,3,4\}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7f34106a67ca254655e40b7786b3e0656a86cba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.475ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{1}=\{\emptyset ,\{1\},\{4\},\{2,3\},\{1,2,3,4\}\}}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{2}=\{\emptyset ,\{1\},\{2\},\{3,4\},\{1,2,3,4\}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc210c18448ccfa2eda2112702d0299f22c25f33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.475ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{2}=\{\emptyset ,\{1\},\{2\},\{3,4\},\{1,2,3,4\}\}}" loading="lazy">.</dd></dl>

Dann ist

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{1}\cap {\mathcal {H}}_{2}=\{\emptyset ,\{1\},\{1,2,3,4\}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/754b7f2c277da861fd2d1e01a0ce3db57b181e1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.836ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{1}\cap {\mathcal {H}}_{2}=\{\emptyset ,\{1\},\{1,2,3,4\}\}}" loading="lazy"></dd></dl>

kein Halbring.

<div class="mw-heading mw-heading3"><h3 id="Produkte_von_Halbringen">Produkte von Halbringen</h3></div>

Definiert man für zwei Mengensysteme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed3e437e9a86b2372c117efc506f112d55466c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.845ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}}_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2bfd4884c6384479f1b003f5325fcea4b2c406e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.845ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M}}_{2}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b95a2e6dbc4b6f2d03b3397691609aa9544230f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Omega _{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/789466adc27ae2abb1feb4e8a00d374d8c6521da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Omega _{2}}" loading="lazy"> das Produkt dieser Mengensysteme als

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{1}\times {\mathcal {M}}_{2}:=\{A\times B\subset \Omega _{1}\times \Omega _{2}\;|\;A\in {\mathcal {M}}_{1},\;B\in {\mathcal {M}}_{2}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a87cb447bc8f0390babd248b20bb9268412bd3ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.846ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{1}\times {\mathcal {M}}_{2}:=\{A\times B\subset \Omega _{1}\times \Omega _{2}\;|\;A\in {\mathcal {M}}_{1},\;B\in {\mathcal {M}}_{2}\}}" loading="lazy">,</dd></dl>

so ist das Produkt von zwei Halbringen wieder ein Halbring. Denn sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{1},{\mathcal {H}}_{2}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0326b15634662df5c8493613aaf496babff001e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.07ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{1},{\mathcal {H}}_{2}}" loading="lazy"> Halbringe und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},B_{1}\in {\mathcal {H}}_{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9a7c21af91e646f6152e1c7242655823a1773e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.508ex; height:2.509ex;" alt="{\displaystyle A_{1},B_{1}\in {\mathcal {H}}_{1}}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2},B_{2}\in {\mathcal {H}}_{2}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76b1007b4ad879cb4be39c9fc6aedc5d52254955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.508ex; height:2.509ex;" alt="{\displaystyle A_{2},B_{2}\in {\mathcal {H}}_{2}}" loading="lazy">, so sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\times A_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1}\times A_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/772aa535e9f4d09ddb92c7ea616ecfbb1103e644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.435ex; height:2.509ex;" alt="{\displaystyle A_{1}\times A_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}\times B_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{1}\times B_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a1364f53a283dd250e2978b13711fa8ba7aadcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.477ex; height:2.509ex;" alt="{\displaystyle B_{1}\times B_{2}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14ffc832d3a6dffa9dbe353e70d8006a475d9247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.876ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}" loading="lazy"> enthalten. Da aber

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<annotation encoding="application/x-tex">{\displaystyle (A_{1}\times A_{2})\cap (B_{1}\times B_{2})=(A_{1}\cap B_{1})\times (A_{2}\cap B_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db928acffddee1ca59dce3f6ec6f79a628bae906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.067ex; height:2.843ex;" alt="{\displaystyle (A_{1}\times A_{2})\cap (B_{1}\times B_{2})=(A_{1}\cap B_{1})\times (A_{2}\cap B_{2})}" loading="lazy"></dd></dl>

gilt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\cap B_{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3c89f4621dd824b5b4827ac23193d492f447e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.198ex; height:2.509ex;" alt="{\displaystyle A_{1}\cap B_{1}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b3de4facc7bd3250b910c4363111ae952f7446f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{1}}" loading="lazy"> liegt und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}\cap B_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{2}\cap B_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14109a6949ec11bd8c3fa5b4e723676fb003d715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.198ex; height:2.509ex;" alt="{\displaystyle A_{2}\cap B_{2}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a40a9c1d9df2f248ea15869968cef102c511684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{2}}" loading="lazy">, ist

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<annotation encoding="application/x-tex">{\displaystyle (A_{1}\times A_{2})\cap (B_{1}\times B_{2})\in {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de413440b0238d0b4e0d6398d3403e405e5fd4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.83ex; height:2.843ex;" alt="{\displaystyle (A_{1}\times A_{2})\cap (B_{1}\times B_{2})\in {\mathcal {H}}_{1}\times {\mathcal {H}}_{2}}" loading="lazy">, das Produkt ist also schnittstabil.

Eine analoge Überlegung unter Verwendung von

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<annotation encoding="application/x-tex">{\displaystyle (A_{1}\times A_{2})\setminus (B_{1}\times B_{2})=\lbrack (A_{1}\setminus B_{1})\times (A_{2}\setminus B_{2})\rbrack \cup \lbrack (A_{1}\setminus B_{1})\times (A_{2}\cap B_{2})\rbrack \cup \lbrack (A_{2}\setminus B_{2})\times (A_{1}\cap B_{1})\rbrack }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b942a4f5ee9b0a36f41351220a4d27c993a8f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.885ex; height:2.843ex;" alt="{\displaystyle (A_{1}\times A_{2})\setminus (B_{1}\times B_{2})=\lbrack (A_{1}\setminus B_{1})\times (A_{2}\setminus B_{2})\rbrack \cup \lbrack (A_{1}\setminus B_{1})\times (A_{2}\cap B_{2})\rbrack \cup \lbrack (A_{2}\setminus B_{2})\times (A_{1}\cap B_{1})\rbrack }" loading="lazy"></dd></dl>

liefert die Differenzeigenschaft eines Halbringes für die Produkte. Beispiel für die Stabilität von Halbringen unter Produktbildung sind die Mengensysteme der halboffenen Intervalle im obigen Beispiel, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}^{2}={\mathcal {I}}\times {\mathcal {I}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}^{2}={\mathcal {I}}\times {\mathcal {I}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9426d0eaa53c448039d355515cf95b99cb6fd27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:11.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {I}}^{2}={\mathcal {I}}\times {\mathcal {I}}}" loading="lazy"> gilt.

Für viele weitere Mengensysteme der Maßtheorie wie Ringe, Algebren und σ-Algebren gilt im Allgemeinen nicht, dass ein Produkt dieser Mengensysteme wieder ein Mengensystem gleicher Art ist. Enthalten Mengensysteme jedoch jeweils einen Halbring, so ist das Produkt stets mindestens ein Halbring. Typisches Beispiel hierfür sind Ringe oder Algebren. Der als Produkt entstehende Halbring wird dann teils als Erzeugendensystem genutzt, um wieder ein Mengensystem mit entsprechender Struktur zu erhalten, das die kartesischen Produkte aller Mengen in den einzelnen Mengensystemen enthaltener Mengen enthält. Beispiel hierfür wäre die <a href="Produkt-%CF%83-Algebra" title="Produkt-σ-Algebra">Produkt-σ-Algebra</a> oder das <a href="Ring_(Mengensystem)#Produkt_von_Ringen" title="Ring (Mengensystem)">hier definierte Produkt von Ringen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{1}\boxtimes {\mathcal {R}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{1}\boxtimes {\mathcal {R}}_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed98ef006388ba107eee45bff871fccebcaa497e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.89ex; height:2.509ex;" alt="{\displaystyle {\mathcal {R}}_{1}\boxtimes {\mathcal {R}}_{2}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Spur_eines_Halbrings">Spur eines Halbrings</h3></div>

Die <a href="Spur_eines_Mengensystems" title="Spur eines Mengensystems">Spur eine Halbrings</a> <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"> bezüglich einer Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, also das Mengensystem

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}|_{U}:=\{A\cap U\;|\;A\in {\mathcal {H}}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4d349fdb6e2e335512e5fffbb455060c2c98e75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.766ex; height:3.009ex;" alt="{\displaystyle {\mathcal {H}}|_{U}:=\{A\cap U\;|\;A\in {\mathcal {H}}\}}" loading="lazy"></dd></dl>

ist immer ein Halbring, unabhängig von der Wahl von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Äquivalente_Definitionen">Äquivalente Definitionen</h2></div>

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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"> sei ein System von Teilmengen 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">. Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"> Mengen sind und wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\triangle B=(A\setminus B)\cup (B\setminus A)}">

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<annotation encoding="application/x-tex">{\displaystyle A\triangle B=(A\setminus B)\cup (B\setminus A)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d41abee74516e83204ae2c2ee449f2ebb5f2befd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.276ex; height:2.843ex;" alt="{\displaystyle A\triangle B=(A\setminus B)\cup (B\setminus A)}" loading="lazy"> die <a href="Symmetrische_Differenz" class="mw-redirect" title="Symmetrische Differenz">symmetrische Differenz</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" loading="lazy"> bezeichnet, dann sind wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcup \emptyset =\emptyset }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc68b0af18c7b9d9958d9f5b92fe4d240bf0d78f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.392ex; height:3.843ex;" alt="{\displaystyle \bigcup \emptyset =\emptyset }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=A\triangle (A\cap B)}">

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<annotation encoding="application/x-tex">{\displaystyle A\setminus B=A\triangle (A\cap B)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8169be96014be5ccca1bf39566c25049d67036a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.508ex; height:2.843ex;" alt="{\displaystyle A\setminus B=A\triangle (A\cap B)}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=A\setminus (A\cap B)}">

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<annotation encoding="application/x-tex">{\displaystyle A\setminus B=A\setminus (A\cap B)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24ee7ab5da0232f500480c470cc42e11339b038a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.637ex; height:2.843ex;" alt="{\displaystyle A\setminus B=A\setminus (A\cap B)}" loading="lazy"> folgende Aussagen <a href="Logische_%C3%84quivalenz" title="Logische Äquivalenz">äquivalent</a>:

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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"> ist ein Mengenhalbring.</li>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70268c47c580acbdb276929f3d8e5cc30318325a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.357ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {H}},\cap )}" loading="lazy"> ist ein <a href="Verband_(Mathematik)" title="Verband (Mathematik)">Halbverband</a> und es gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04f1d69b2be8936d74fd6ede6d1cd0aa1cbd9f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.314ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow }" loading="lazy"> Es gibt paarweise disjunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5702ad41888e71619b500869b7a6a02793a07b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.173ex; height:2.509ex;" alt="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}">

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<annotation encoding="application/x-tex">{\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/114dae1d3b14a568766f32646005d534fadde832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.932ex; height:2.843ex;" alt="{\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}" loading="lazy"></li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06d73d359ffb710f590e1ec9a60bda09b382d9a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.967ex; height:2.509ex;" alt="{\displaystyle \emptyset \in {\mathcal {H}}}" loading="lazy"> und es gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}">

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<annotation encoding="application/x-tex">{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04a9f5d393a2a211ee9b3ab81201176cbf9901df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.854ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}" loading="lazy"> und es existiert ein endliches Teilsystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}},}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bd8f31a64d7877958a6ba802006ddda61c1661e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}},}" loading="lazy"> dessen Elemente paarweise disjunkt sind, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=\bigcup {\mathcal {C}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42c65edf0807939744cab0e6c371b07d436f6c6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.008ex; height:3.843ex;" alt="{\displaystyle A\setminus B=\bigcup {\mathcal {C}}}" loading="lazy">. <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"> kann hierbei auch leer sein.</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d2c6b8f2aa2575c3360f9b42a1b4208cc7a5d49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.225ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}\neq \emptyset }" loading="lazy"> und es gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}">

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<annotation encoding="application/x-tex">{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04a9f5d393a2a211ee9b3ab81201176cbf9901df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.854ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}" loading="lazy"> und es gibt paarweise disjunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5702ad41888e71619b500869b7a6a02793a07b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.173ex; height:2.509ex;" alt="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\triangle B=C_{1}\cup \dotsb \cup C_{n}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\triangle B=C_{1}\cup \dotsb \cup C_{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eadddf1d5594dcaeb65bd51e175c92544682a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.803ex; height:2.509ex;" alt="{\displaystyle A\triangle B=C_{1}\cup \dotsb \cup C_{n}.}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d2c6b8f2aa2575c3360f9b42a1b4208cc7a5d49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.225ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}\neq \emptyset }" loading="lazy"> und es gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04a9f5d393a2a211ee9b3ab81201176cbf9901df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.854ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {H}}\Rightarrow A\cap B\in {\mathcal {H}}}" loading="lazy"> und falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb8124cb68686ede7083aa2a5a821f262eb62954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle B\subseteq A}" loading="lazy"> gilt, gibt es paarweise disjunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5702ad41888e71619b500869b7a6a02793a07b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.173ex; height:2.509ex;" alt="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/114dae1d3b14a568766f32646005d534fadde832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.932ex; height:2.843ex;" alt="{\displaystyle A\setminus B=C_{1}\cup \dotsb \cup C_{n}.}" loading="lazy"></li></ul>

Außerdem ergibt sich induktiv:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5702ad41888e71619b500869b7a6a02793a07b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.173ex; height:2.509ex;" alt="{\displaystyle C_{1},\dotsc ,C_{n}\in {\mathcal {H}},n\in \mathbb {N} ,}" loading="lazy"> sind paarweise disjunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow C_{1}\cup \dots \cup C_{n}=C_{1}\triangle \dotsb \triangle C_{n}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Rightarrow C_{1}\cup \dots \cup C_{n}=C_{1}\triangle \dotsb \triangle C_{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8a08391fdcbf1421fcf3d6ebe847f115b98c1f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.425ex; height:2.509ex;" alt="{\displaystyle \Rightarrow C_{1}\cup \dots \cup C_{n}=C_{1}\triangle \dotsb \triangle C_{n}.}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Halbringe_im_engeren_Sinne">Halbringe im engeren Sinne</h2></div>

Manche Autoren nennen das oben definierte Mengensystem einen Semiring/Halbring im weiteren Sinne (i. w. S.) und definieren noch einen Semiring/Halbring im engeren Sinne (i. e. S.) als eine Mengensystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a1cc103563219127f59aec7ed9327a3595566dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}" loading="lazy">,<a href="#cite_note-3">[3]</a>

<ol><li>das die leere Menge enthält,</li>

<li>das schnittstabil ist,</li>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66417283191a8ab2f010ea32e33e524a962fc354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.787ex; height:2.509ex;" alt="{\displaystyle A,B\in {\mathcal {W}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> existiert, sodass paarweise disjunkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},\dotsc ,C_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1},\dotsc ,C_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4f9836d558f83f81fe9c7db9c6ce14148e22680.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.775ex; height:2.509ex;" alt="{\displaystyle C_{1},\dotsc ,C_{n}}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a1cc103563219127f59aec7ed9327a3595566dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}" loading="lazy"> existieren, für die</li></ol>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\setminus A=\bigcup _{i=1}^{n}C_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7418523b1f6c4232af90d3b9b5e45f2ecd3556e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.317ex; height:6.843ex;" alt="{\displaystyle B\setminus A=\bigcup _{i=1}^{n}C_{i}}" loading="lazy"></dd></dl></dd></dl></dd></dl>

<dl><dd><dl><dd>gilt und zusätzlich

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\cup \bigcup _{i=1}^{k}C_{i}\in {\mathcal {W}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbe992ea4e010e7a930c9d2b46e682d9062306f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.088ex; height:7.343ex;" alt="{\displaystyle A\cup \bigcup _{i=1}^{k}C_{i}\in {\mathcal {W}}}" loading="lazy"></dd></dl></dd></dl></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \{1,\dots ,n\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0b7dcc443b9db3f49d5f3fd0c0ae7120885586b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.112ex; height:2.843ex;" alt="{\displaystyle k\in \{1,\dots ,n\}}" loading="lazy">.</dd></dl></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Verwandte_Mengensysteme">Verwandte Mengensysteme</h2></div>

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

Jeder <a href="Ring_(Mengensystem)" title="Ring (Mengensystem)">Mengenring</a> ist ein Mengenhalbring, jedoch ist nicht jeder Mengenhalbring ein Mengenring: Über der Grundmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\{0,1,2,3,4\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{0,1,2,3,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe1f3e93b4c9c7c14736756c95ca9bd6dd87e463.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.049ex; height:2.843ex;" alt="{\displaystyle \Omega =\{0,1,2,3,4\}}" loading="lazy"> ist das Mengensystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4916f1cd60ee49dea2570acba33700b8254166f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.028ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}" loading="lazy"> ein Halbring, aber kein Mengenring, da es nicht differenzstabil ist.

Verwendet man einen 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"> als Erzeuger eines Ringes, so hat der <a href="Erzeugter_Ring" class="mw-redirect" title="Erzeugter Ring">erzeugte Ring</a> die Form

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mtext> paarweise disjunkt</mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}=\left\{\left.\bigcup _{j=1}^{n}A_{j}\,\right|\,A_{1},\dotsc ,A_{n}\in {\mathcal {H}},\quad A_{j}{\text{ paarweise disjunkt}}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fd95035720ce90f821fc508f819a606c70510e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.473ex; height:7.843ex;" alt="{\displaystyle {\mathcal {R}}=\left\{\left.\bigcup _{j=1}^{n}A_{j}\,\right|\,A_{1},\dotsc ,A_{n}\in {\mathcal {H}},\quad A_{j}{\text{ paarweise disjunkt}}\right\}}" loading="lazy">.</dd></dl>

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

Per Definition ist jeder Halbring (im engeren Sinn / im weiteren Sinn) genau dann eine <a href="Semialgebra" title="Semialgebra">Semialgebra</a> (im engeren Sinn / im weiteren Sinn), wenn er die Obermenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> enthält. Beispiel für einen Halbring, der keine Semialgebra ist, wäre somit der Halbring

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4916f1cd60ee49dea2570acba33700b8254166f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.028ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2,3\}\}}" loading="lazy"></dd></dl>

auf der Grundmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\{0,1,2,3,4\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{0,1,2,3,4\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe1f3e93b4c9c7c14736756c95ca9bd6dd87e463.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.049ex; height:2.843ex;" alt="{\displaystyle \Omega =\{0,1,2,3,4\}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Weitere_Mengensysteme">Weitere Mengensysteme</h3></div>

Da jeder Mengenring ein Halbring ist, sind <a href="Algebra_(Mengensystem)" title="Algebra (Mengensystem)">Mengenalgebren</a>, <a href="%CE%A3-Ring" title="Σ-Ring">σ-Ringe</a>, <a href="%CE%94-Ring_(Mengensystem)" title="Δ-Ring (Mengensystem)">δ-Ringe</a> und <a href="%CE%A3-Algebra" title="Σ-Algebra">σ-Algebren</a> immer auch Halbringe, da sie alle auch Ringe sind. Die Umkehrung gilt im Allgemeinen nicht, wie das obige Beispiel zeigt.

<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>:<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 class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Halbring+%28Mengensystem%29&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%2FHeidelberg&rft.pub=Springer-Verlag" style="display:none"> </li>

<li><a href="Ernst_Henze_(Mathematiker)" title="Ernst Henze (Mathematiker)">Ernst Henze</a>: Einführung in die Maßtheorie. 2., überarbeitete Auflage. Bibliographisches Institut, Mannheim u. a. 1985, ISBN 3-411-03102-6.</li>

<li><a rel="nofollow" class="external text" href="http://www.probability.net/WEBcaratheodory.pdf">Caratheodory’s Extension.</a> Skript bei: probability.net. (PDF-Datei; 115 kB).</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Elstrodt: Maß- und Integrationstheorie. 2009, S. 20.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Elstrodt: Maß- und Integrationstheorie. 2009, S. 20.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Norbert Kusolitsch: <cite style="font-style:italic">Maß- und Wahrscheinlichkeitstheorie</cite>. Eine Einführung. 2., überarbeitete und erweiterte Auflage. Springer-Verlag, Berlin/Heidelberg 2014, ISBN 978-3-642-45386-1, S. 13, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-45387-8">10.1007/978-3-642-45387-8</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Halbring+%28Mengensystem%29&rft.au=Norbert+Kusolitsch&rft.btitle=Ma%C3%9F-+und+Wahrscheinlichkeitstheorie&rft.date=2014&rft.doi=10.1007%2F978-3-642-45387-8&rft.edition=2.%2C+%C3%BCberarbeitete+und+erweiterte&rft.genre=book&rft.isbn=9783642453861&rft.pages=13&rft.place=Berlin%2FHeidelberg&rft.pub=Springer-Verlag" style="display:none">

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