Pi system

In mathematics, a π-system on a set Ω is a set P, consisting 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.

Properties

• A σ-algebra is necessarily a π-system.
• A π-system which is also a Dynkin system is a σ-algebra.
• A topology is a π-system.

Uniqueness of Extension

A finite measure may be considered to be uniquely determined by its values on a π-system in the following sense. Let μ and υ be measures on (X, Σ) and be a π-system that generates Σ. If

then μ=υ.

See also

• Dynkin's lemma

References

• Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN 0-387-22833-0. 
• David Williams (2007). Probability with Martingales. Cambridge University Press. ISBN 0521406056.