Analytic Borel space

Also: analytic measurable space

2020 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A0554H05 [MSN][ZBL]

$ \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 is called analytic if it is countably separated and isomorphic to a quotient space of a standard Borel space.

See below for an equivalent definition.

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 (or just countably separated measurable) 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 given 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.

See [1, Sect. 25.A], [2, Sect. 4] for these, and some other, definitions of analytic sets and spaces.

Measurable injections

Several results on standard Borel spaces generalize to analytic Borel spaces (see [M], Sect. 4; [S], Sect. 4.5).

Theorem 1a. If a bijective map between analytic Borel spaces is measurable then the inverse map is also measurable.

Theorem 1b. If σ-algebras $\A$, $\B$ on $X$ are such that $\A\subset\B$ and $(X,\A)$, $(X,\B)$ are analytic then $\A=\B$.

Example. The real line with the Lebesgue σ-algebra is not analytic (by Theorem 1b).

Theorem 2a. If a bijective map from an analytic Borel space to a countably separated measurable space is measurable then the inverse map is also measurable.

Theorem 2b. If σ-algebras $\A$, $\B$ on $X$ are such that $\A\subset\B$, $(X,\A)$ is countably separated and $(X,\B)$ is analytic then $\A=\B$.

Theorem 2c. If $(X,\A)$ is an analytic Borel space then $\A$ is generated by every at most countable separating subset of $\A$.

The Blackwell-Mackey theorem generalizes readily from standard to analytic spaces, since a quotient space of an analytic space evidently is also a quotient space of a standard space.

References

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