Universally Measurable Set - Finiteness Condition

Finiteness Condition

The condition that the measure be a probability measure; that is, that the measure of itself be 1, is less restrictive than it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say, N₀=[0,1), N₁=[1,2), N₂=[-1,0), N₃=[2,3), N₄=[-2,-1), and so on. Now letting μ be Lebesgue measure, define a new measure ν by

Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable. More generally a universally measurable set must be measurable with respect to every sigma-finite measure that measures all Borel sets.