User:Boris Tsirelson/sandbox1

$\newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ The term "universally measurable" may be applied to

• a measurable space;
• a subset of a measurable space;
• a metric space.

Let $(X,\A)$ be a measurable space. A subset $A\subset X$ is called universally measurable if it is $\mu$-measurable for every finite measure $\mu$ on $(X,\A)$. In other words: $\mu_*(A)=\mu^*(A)$ where $\mu_*,\mu^*$ are the inner and outer measures for $\mu$, that is,

$ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$

(See [S, p. 170].)

A separable metric space is called universally measurable if it is a universally measurable subset (as defined above) of its completion. Here the completion, endowed with the Borel σ-algebra, is treated as a measurable space. (See [S, p. 170], [D, Sect. 11.5].)

A measurable space is called universally measurable if it is isomorphic to some universally measurable metric space (as defined above) with the Borel σ-algebra. (See [S, p. 171].)

Theorem 1 (Shortt). A countably generated separated measurable space $(X,\A)$ is universally measurable if and only if for every finite measure $\mu$ on $(X,\A)$ there exists a subset $A\in\A$ of full measure (that is, $\mu(X\setminus A)=0$) such that $A$ (treated as a subspace) is itself a standard Borel space. ([S, Lemma 4])

Theorem 2 (Shortt). The following two conditions on a separable metric space are equivalent:

(a) it is a universally measurable metric space;

(b) the corresponding measurable space (with the Borel σ-algebra) is universally measurable.

Evidently, (a) implies (b); surprisingly, also (b) implies (a), which involves a Borel isomorphism (rather than isometry or homeomorphism) between two metric spaces.

References

[S]	Rae M. Shortt, "Universally measurable spaces: an invariance theorem and diverse characterizations", Fundamenta Mathematicae 121 (1984), 169–176.   MR0765332   Zbl 0573.28018
[P]	David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).   MR1873379   Zbl 0992.60001
[K]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[BK]	Howard Becker and Alexander S. Kechris, "The descriptive set theory of Polish group actions", Cambridge (1996).   MR1425877   Zbl 0949.54052
[D]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
[M]	George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[H]	Paul R. Halmos, "Measure theory", v. Nostrand (1950).   MR0033869   Zbl 0040.16802
[R]	Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953).   MR0055409   Zbl 0052.05301