Standard Borel space

Also: standard measurable space

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

$ \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.

The isomorphism theorem

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 ("the isomorphism theorem", see [1, Sect. 15.B]) 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.

Measurable injections

Here is another important fact (see [3, Th. 3.2] or [1, Sect. 15.A]) 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).

Recall a topological fact similar to Theorem 2: if a bijective map between compact Hausdorff topological spaces is continuous then the inverse map is also continuous. Moreover, if a Hausdorff topology is weaker than a compact topology then these two topologies are equal, which has the following Borel-space counterpart stronger than Theorem 2 (in three equivalent forms).

Theorem 3a. If a bijective map from a standard Borel space to a countably separated measurable space is measurable then the inverse map is also measurable.

Theorem 3b. If σ-algebras $\A$, $\B$ on $X$ are such that $\A\subset\B$, $(X,\A)$ is countably separated and $(X,\B)$ is standard then $\A=\B$.

Theorem 3c. If $(X,\A)$ is a standard Borel space then $\A$ is generated by every at most countable separating subset of $\A$. (See [3, Sect. 3].

If a subset of a Hausdorff topological space is itself a compact topological space then it is a compact subset, which also has a Borel-space counterpart.

Theorem 4. If a subset of a countably separated measurable space is itself a standard Borel space then it is a measurable subset.

The analogy breaks down for maps that are not one-to-one. A continuous image of a compact topological space is always a compact set, in contrast to the following.

Fact. If $(X,\A)$ and $(Y,\B)$ are standard Borel spaces and $f:X\to Y$ is a measurable map then $f(X)$ is not necessarily measurable.

That is, the set $f(X)$ need not belong to $\B$. It is a so-called analytic set, and it is universally measurable.

For one-to-one maps a positive result is available (follows easily from Theorems 3b and 4).

Theorem 5. If $(X,\B)$ is a standard Borel space, $(Y,\A)$ a countably separated measurable space, and $f:X\to Y$ a measurable one-to-one map then $f(X)$ is measurable.

Blackwell-Mackey theorem

On the other hand, Theorem 3 has a counterpart for many-to-one maps. (See [4, Sect. 4.5].) First, note that an arbitrary map $f:X\to Y$ is a composition of the projection $p:X\to X/f$ and a one-to-one map $g:X/f\to Y$; here $X/f=\{f^{-1}(y):y\in f(X)\}$ (the quotient set) and $p(x)=f^{-1}(f(x))$ (the equivalence class of $x$). If in addition $X,Y$ are measurable spaces and $f$ a measurable map then $p$ and $g$ are measurable. (Here $X/f$ is treated as a quotient measurable space.)

Theorem 6. Let $(X,\B)$ be a standard Borel space, $(Y,\A)$ a countably separated measurable space, $f:X\to Y$ a measurable map, $f(X)=Y$, and $p:X\to X/f$, $g:X/f\to Y$ as above. Then $g^{-1}$ is measurable.

Reformulating it in terms of the quotient space one generalizes Theorem 3 as follows.

Theorem 7a. If a bijective map from a quotient space of a standard Borel space to a countably separated measurable space is measurable then the inverse map is also measurable.

Theorem 7b. If σ-algebras $\A$, $\B$ on $X$ are such that $\A\subset\B$, $(X,\A)$ is countably separated and $(X,\B)$ is a quotient space of a standard Borel space then $\A=\B$.

Theorem 7c. If $(X,\A)$ is a quotient space of a standard Borel space then $\A$ is generated by every at most countable separating subset of $\A$. (Of course, the conclusion is void unless $(X,\A)$ is countably separated.)

A countably separated quotient space of a standard Borel space is called analytic Borel space.

Measurable graphs

The graph $\{(x,f(x)):x\in X\}$ of a map $f:X\to Y$ is a subset of $X\times Y$. Generally, measurability of the graph is necessary (under mild conditions) but not sufficient for measurability of the map. But for standard spaces it is also sufficient. (See [1, Sect. 14.C]. The sufficiency follows easily from Theorem 5. Also, Theorem 3a follows easily from Theorem 8 below.)

Theorem 8. If $(X,\A)$ and $(Y,\B)$ are standard Borel spaces and $f:X\to Y$ then measurability of $f$ is equivalent to measurability of the graph of $f$.

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
[4]	S.M. Srivastava, "A course on Borel sets", Springer-Verlag (1998).   MR1619545  Zbl 0903.28001