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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 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. ==Examples== * , the intervals form a π-system, and the intervals form a π-system, if the empty set is also included. * The topology (collection of open subsets) of any topological space is a π-system. * For any collection Σ of subsets of Ω, there exists a π-system which is the unique smallest π-system of Ω to contain every element of Σ, and is called the π-system '' generated '' by Σ. * For any measurable function , the set defines a π-system, and is called the π-system '' generated '' by ''f''. (Alternatively, 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「pi system」の詳細全文を読む スポンサード リンク
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