Pi-system (original) (raw)
Ein π-System, auch durchschnittstabiles Mengensystem oder kurz schnittstabiles System genannt, ist ein spezielles Mengensystem, das im axiomatischen Aufbau der Wahrscheinlichkeitstheorie und der Maßtheorie verwendet werden kann.
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| dbo:abstract | Ein π-System, auch durchschnittstabiles Mengensystem oder kurz schnittstabiles System genannt, ist ein spezielles Mengensystem, das im axiomatischen Aufbau der Wahrscheinlichkeitstheorie und der Maßtheorie verwendet werden kann. (de) In mathematics, a π-system (or pi-system) on a set is a collection of certain subsets of such that * is non-empty. * If then That is, is a non-empty family of subsets of that is closed under non-empty finite intersections.The importance of π-systems arises from the fact that if two probability measures agree on a π-system, then they agree on the 𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra as well. This is the case whenever the collection of subsets for which the property holds is a 𝜆-system. π-systems are also useful for checking independence of random variables. This is desirable because in practice, π-systems are often simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many sets So instead we may examine the union of all 𝜎-algebras generated by finitely many sets This forms a π-system that generates the desired 𝜎-algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a π-system that generates the very important Borel 𝜎-algebra of subsets of the real line. (en) En mathématiques, un π-système (ou pi-système) sur un ensemble est un ensemble de parties de stable par intersection. Les π-systèmes font partie des familles d'ensembles que l'on rencontre en théorie de la mesure et théorie des probabilités. On sait par exemple grâce au lemme de classe monotone que deux mesures finies, et en particulier deux mesures de probabilités, dont les valeurs coïncident sur un π-système, coïncident également sur la tribu engendrée par le dit π-système. Les π-systèmes offrent donc une famille d'ensembles de prédilection, et relativement simple, pour vérifier l'égalité de deux mesures ou bien l'unicité de la construction d'une mesure. (fr) In matematica, un sistema pi, o anche -sistema, su un insieme è una famiglia P non vuota di sottoinsiemi di (ovvero ), tale che l'intersezione di due elementi (e quindi di un numero finito di elementi) di P è ancora in P; ovvero P è stabile per intersezioni finite. (it) π-układ – rodzina zbiorów zamknięta na branie skończonych przekrojów, mająca zastosowanie przede wszystkim w teorii mnogości, teorii miary i rachunku prawdopodobieństwa. (pl) Em um conjunto Ω, em matemática, um sistema Pi (ou sistema π) é um conjunto P de determinados subconjuntos de Ω, tal que: * P não é vazio. * A ∩ B ∈ P, sempre que A e B estão em P. (pt) |
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| rdfs:comment | Ein π-System, auch durchschnittstabiles Mengensystem oder kurz schnittstabiles System genannt, ist ein spezielles Mengensystem, das im axiomatischen Aufbau der Wahrscheinlichkeitstheorie und der Maßtheorie verwendet werden kann. (de) En mathématiques, un π-système (ou pi-système) sur un ensemble est un ensemble de parties de stable par intersection. Les π-systèmes font partie des familles d'ensembles que l'on rencontre en théorie de la mesure et théorie des probabilités. On sait par exemple grâce au lemme de classe monotone que deux mesures finies, et en particulier deux mesures de probabilités, dont les valeurs coïncident sur un π-système, coïncident également sur la tribu engendrée par le dit π-système. Les π-systèmes offrent donc une famille d'ensembles de prédilection, et relativement simple, pour vérifier l'égalité de deux mesures ou bien l'unicité de la construction d'une mesure. (fr) In matematica, un sistema pi, o anche -sistema, su un insieme è una famiglia P non vuota di sottoinsiemi di (ovvero ), tale che l'intersezione di due elementi (e quindi di un numero finito di elementi) di P è ancora in P; ovvero P è stabile per intersezioni finite. (it) π-układ – rodzina zbiorów zamknięta na branie skończonych przekrojów, mająca zastosowanie przede wszystkim w teorii mnogości, teorii miary i rachunku prawdopodobieństwa. (pl) Em um conjunto Ω, em matemática, um sistema Pi (ou sistema π) é um conjunto P de determinados subconjuntos de Ω, tal que: * P não é vazio. * A ∩ B ∈ P, sempre que A e B estão em P. (pt) In mathematics, a π-system (or pi-system) on a set is a collection of certain subsets of such that * is non-empty. * If then That is, is a non-empty family of subsets of that is closed under non-empty finite intersections.The importance of π-systems arises from the fact that if two probability measures agree on a π-system, then they agree on the 𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra as well. This is the case whenever the collection of subsets for which the property holds is a 𝜆-system. π-systems are also useful for checking independence of random variables. (en) |
| rdfs:label | Π-System (de) Pi-système (fr) Sistema pi (it) Pi-system (en) Π-układ (pl) Sistema Pi (pt) |
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