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- Pi_system abstract "In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that P is non-empty. A ∩ B ∈ P whenever A and B are in P.That is, P is a non-empty family of subsets of Ω that is closed under finite intersections.The importance of π-systems arise 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 random sets . This forms a π-system that generates the desired σ-algebra. Another example is the collection of all interval subsets 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.".
- Pi_system wikiPageID "4050532".
- Pi_system wikiPageRevisionID "599333947".
- Pi_system hasPhotoCollection Pi_system.
- Pi_system subject Category:Measure_theory.
- Pi_system subject Category:Set_families.
- Pi_system type Abstraction100002137.
- Pi_system type Family108078020.
- Pi_system type Group100031264.
- Pi_system type Organization108008335.
- Pi_system type SetFamilies.
- Pi_system type SocialGroup107950920.
- Pi_system type Unit108189659.
- Pi_system type YagoLegalActor.
- Pi_system type YagoLegalActorGeo.
- Pi_system type YagoPermanentlyLocatedEntity.
- Pi_system comment "In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that P is non-empty. A ∩ B ∈ P whenever A and B are in P.That is, P is a non-empty family of subsets of Ω that is closed under finite intersections.The importance of π-systems arise 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.".
- Pi_system label "Pi system".
- Pi_system label "Sistema pi".
- Pi_system label "Π-układ".
- Pi_system sameAs Sistema_pi.
- Pi_system sameAs Π-układ.
- Pi_system sameAs m.0bfkvq.
- Pi_system sameAs Q3962249.
- Pi_system sameAs Q3962249.
- Pi_system sameAs Pi_system.
- Pi_system wasDerivedFrom Pi_system?oldid=599333947.
- Pi_system isPrimaryTopicOf Pi_system.