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.

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 $mathcal\{I\}$ be a π-system that generates Σ. If

*

* $mu\; |\_\{mathcal\{I\; =\; u\; |\_\{mathcal\{I$

then μ=υ.

ee also

* Dynkin's lemma

References

* cite book
last = Gut
first = Allan
title = Probability: A Graduate Course
publisher = Springer
year = 2005
location = New York
doi = 10.1007/b138932
isbn = 0-387-22833-0
*