any σ-finite measure is equivalent to a probability measure
The following theorem states that for any σ-finite (http://planetmath.org/SigmaFinite) measure μ, there is an equivalent
probability measure ℙ — that is, the sets A satisfying μ(A)=0 are the same as those satisfying ℙ(A)=0.
This result allows statements about probability measures to be generalized to arbitrary σ-finite measures.
Theorem.
Any nonzero σ-finite measure μ on a measurable space (X,A) is equivalent to a probability measure P on (X,A). In particular, there is a positive measurable function
f:X→(0,∞) satisfying ∫f𝑑μ=1, and P(A)=∫Af𝑑μ for all A∈A.
Proof.
Title | any σ-finite measure is equivalent to a probability measure |
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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 |