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. * If A and B are in P then A ∩ B ∈ P. That is, P is a non-empty family of subsets of Ω that is closed under 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 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.

Technology Types

measure theoryset family

Synonyms

Dynkin's π-system

Translations

Pi-système (fr)Sistema pi (it)Sistema Pi (pt)Π-System (de)Π-układ (pl)