0.1 Pi System
A \(\pi\)-system on a set \(\Omega\) is a collection \(P\) of certain subsets of \(\Omega\) such that 1. \(P\) is non-empty; and 2. If \(A,B\in P\) then \(A \cap B \in P\).
That is, \(P\) is a non-empty family of subsets of \(\Omega\) that is closed under non-empty finite intersections.
The importance of \(\pi\)-systems arises from the fact that if two [[probability-measure|probability measures]] agree on a \(\pi\)-system, then they agree on the [[generated-sigma-algebra|\(\sigma\)-algebra generated]] by that \(\pi\)-system. Moreover, if other properties, such as equality of integrals, hold for the \(\pi\)-system, then they hold for the [[generated-sigma-algebra|generated \(\sigma\)-algebra]] as well. This is the case whenever the collection of subsets for which the property holds is a Lambda System.