uniqueness of measures extended from a π-system

The following theorem allows measuresMathworldPlanetmath to be uniquely defined by specifying their values on a π-system (http://planetmath.org/PiSystem) instead of having to specify the measure of every possible measurable setMathworldPlanetmath. For example, the collectionMathworldPlanetmath of open intervals (a,b) forms a π-system generating the Borel σ-algebra (http://planetmath.org/BorelSigmaAlgebra) and consequently the Lebesgue measureMathworldPlanetmath μ is uniquely defined by the equality μ((a,b))=b-a.


Let λ, μ be measures on a measurable spaceMathworldPlanetmath (X,A). Suppose that A is a π-system on X generating A such that λ=μ on A and that there exists a sequence SnA with n=1Sn=X and λ(Sn)<. Then, λ=μ.


Choose any TA such that λ(T)< and set ={S𝒜:λ(ST)=μ(ST)}. For any SA, STA and the requirement that λ,μ agree on A gives S, so contains A. We show that is a Dynkin system in order to apply Dynkin’s lemma. It is clear that X. Suppose that S1S2 are in . Then, the additivity of λ and μ gives


and therefore S2S1. Now suppose that Sn is an increasing sequence of sets in increasing to SX. Then, monotone convergence of λ and μ gives


so S and is a Dynkin system containing A. By Dynkin’s lemma this shows that contains σ(A)=𝒜.

We have shown that λ(ST)=μ(ST) for any S𝒜 and TA with λ(T)<. In the particular case where XA and λ,μ are finite measures then it follows that λ(S)=μ(S) simply by taking T=X. More generally, choose a sequence of sets TnA satisfying λ(Tn)< and nTn=X. For any S𝒜, Sn(STn)m=1n-1Tm is a pairwise disjoint sequence of sets in 𝒜 with SnTn and nSn=S. So, λ(Sn)=μ(Sn) and the countable additivity of λ and μ gives


Title uniqueness of measures extended from a π-system
Canonical name UniquenessOfMeasuresExtendedFromApisystem
Date of creation 2013-03-22 18:33:08
Last modified on 2013-03-22 18:33:08
Owner gel (22282)
Last modified by gel (22282)
Numerical id 8
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Related topic LebesgueMeasure
Related topic DynkinsLemma