any σ-finite measure is equivalent to a probability measure

The following theorem states that for any σ-finite ( measureMathworldPlanetmath μ, there is an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath probability measure — that is, the sets A satisfying μ(A)=0 are the same as those satisfying (A)=0. This result allows statements about probability measures to be generalized to arbitrary σ-finite measures.


Any nonzero σ-finite measure μ on a measurable spaceMathworldPlanetmathPlanetmath (X,A) is equivalent to a probability measure P on (X,A). In particular, there is a positive measurable functionMathworldPlanetmath f:X(0,) satisfying f𝑑μ=1, and P(A)=Af𝑑μ for all AA.


Let A1,A2, be a sequence in 𝒜 such that μ(Ak)< and kAk=X. Then it is easily verified that


satisfies 1g>0 and g𝑑μ<. So, setting f=g/g𝑑μ, we have f𝑑μ=1 and therefore (A)Af𝑑μ is a probability measure equivalent to μ. ∎

Title any σ-finite measure is equivalent to a probability measure
Canonical name AnysigmafiniteMeasureIsEquivalentToAProbabilityMeasure
Date of creation 2013-03-22 18:33:44
Last modified on 2013-03-22 18:33:44
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Classification msc 28A10
Related topic SigmaFinite
Related topic Measure