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. 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 etc., endowed with their Borel σ-algebras. That is instructive: topological notions such as dimension, connectedness etc. do not apply to Borel spaces.

References

[3]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995) | MR1321597 | Zbl 0819.04002
[5]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) | MR0982264 | Zbl 0686.60001