Standard Borel space

Also: standard measurable space

2020 Mathematics Subject Classification: Primary: 28A05 Secondary: 03E1554H05 [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 $(X,\A)$ is called standard if it satisfies the following equivalent conditions:

• $(X,\A)$ is isomorphic to some compact metric space with the Borel σ-algebra;
• $(X,\A)$ is isomorphic to some separable complete metric space with the Borel σ-algebra;
• $(X,\A)$ is isomorphic to some Borel subset of some separable complete metric space with the Borel σ-algebra.

Finite and countable standard Borel spaces are trivial: all subsets are measurable. Two such spaces are isomorphic if and only if they have the same cardinality, which is trivial. But the following result is surprising and highly nontrivial.

Theorem 1. All uncountable standard Borel spaces are mutually isomorphic.

That is, up to isomorphism we have "the" uncountable standard Borel space. Its "incarnations" include $\R^n$ (for every $n\ge1$), separable Hilbert spaces, the Cantor set, the set of all irrational numbers etc. (these are separable complete metric spaces or Borel sets in such spaces), endowed with their Borel σ-algebras. That is instructive: topological notions such as dimension, connectedness, compactness etc. do not apply to Borel spaces.

Here is another important fact in two equivalent forms.

Theorem 2a. If a bijective map between standard Borel spaces is measurable then the inverse map is also measurable.

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

Example. The real line with the Lebesgue σ-algebra is not standard (by Theorem 2b).

References

[1]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995) | MR1321597 | Zbl 0819.04002
[2]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) | MR0982264 | Zbl 0686.60001
[3]	George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165 | MR0089999 | Zbl 0082.11201