Pi system

In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that

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.