Universally measurable set について

. In particular, a universally measurable set of reals is necessarily Lebesgue measurable (see #Finiteness condition) below.
Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.
==Finiteness condition==
The condition that the measure be a probability measure; that is, that the measure of

X

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, ''N''₀=[0,1), ''N''₁=[1,2), ''N''₂=[-1,0), ''N''₃=[2,3), ''N''₄=[-2,-1), and so on. Now letting μ be Lebesgue measure, define a new measure ν by
:

\nu(A)=\sum_^\infty \frac{2^{n+1}}\mu(A\cap N_i)

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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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