User:Boris Tsirelson/sandbox1

Also: analytic measurable space

Category:Descriptive set theory Category:Classical measure theory

[ 2010 Mathematics Subject Classification MSN: 03E15,(28A05,54H05) | MSCwiki: 03E15   + 28A05,54H05  ]

$ \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\P}{\mathbf P} $ A Borel space $(X,\A)$ is called analytic if it is countably separated and isomorphic to a quotient space of a standard Borel space.

This is one out of several equivalent definitions (see below).

Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable analytic Borel spaces are of cardinality continuum. Some, but not all, of them are standard; these are mutually isomorphic. Some additional (to ZFC) set-theoretic axioms imply that all nonstandard analytic Borel spaces are mutually isomorphic (see [1, Sect. 26.D]).

Non-example. The quotient group $\R/\Q$ (real numbers modulo rational numbers, additive) may be thought of as a quotient measurable space, $\R$ being endowed with its Borel σ-algebra. Then $\R/\Q$ is a quotient space of a standard Borel space, but not an analytic Borel space, because it is not countably separated. (See [2, Sect. 5].)

Relations to analytic sets

A subset of a standard Borel space is called analytic if it is the image of a standard Borel space under a Borel map. (See [1, Sect. 14.A].)

As every subset of a measurable space, an analytic set is itself a measurable space (a subspace of the standard Borel space).

Definition 2 (equivalent). A Borel space is called analytic if it is isomorphic to an analytic set.

Equivalence of the two definitions follows from the Blackwell-Mackey theorem and the following simple fact: every countably separated measurable space admits a one-to-one measurable map to a standard Borel space.

---

Theorem 1a. If a bijective map between analytic sets is measurable then the inverse map is also measurable. (See [3, Sect. 4.5].)

References

[1]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597  Zbl 0819.04002
[2]	George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[3]	S.M. Srivastava, "A course on Borel sets", Springer-Verlag (1998).   MR1619545  Zbl 0903.28001