Standard Borel space

Also: standard measurable space

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

$\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.

Basic constructions and standardness

The product of two standard Borel spaces is a standard Borel space. The same holds for countably many factors. (For uncountably many factors of at least two points each, the product is not countably separated, therefore not standard.)

A measurable subset of a standard Borel space, treated as a subspace, is a standard Borel space. (It never happens to a nonmeasurable subset, see Theorem 4 below.)

The disjoint union of two standard Borel spaces is a standard Borel space.

(See [K, Sect. 12.B].)

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 [K, 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 [M, Th. 3.2] or [K, 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 [M, Sect. 3].)

If a subset of a Hausdorff topological space is itself a compact topological space then it is a closed 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 [S, 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 [K, 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 $f$.

Relations to measures

Let $(X,\A)$ be a standard Borel space and $\mu:\A\to[0,1]$ a probability measure. Then the completion of $(X,\A,\mu)$ is a standard probability space. (See [K, Sect. 17.F].)

Let $(X,\A)$ be a standard Borel space, $M$ the set of all probability measures on $(X,\A)$, and $\B$ the σ-algebra on $M$ generated by the evaluation maps $\mu\mapsto\mu(A)$ for all $A\in\A$ (or equivalently, by the maps $\mu\mapsto\int f\,\rd\mu$ for all bounded measurable functions $f:X\to\R$). Then $(M,\B)$ is a standard Borel space. The same holds for measures $\mu:\A\to[0,\infty)$, signed measures $\mu:\A\to\R$, complex-valued measures etc., as far as these measures are of finite (total) variation. (See [K, Sect. 17.E].)

Space of closed sets

Let $T$ be a complete separable metric space, and $\F(T)$ the Effros Borel space of closed subsets of $T$.

Then $\F(T)$ is a standard Borel space.

The set of all compact subsets of $T$ is a Borel subset of $\F(T)$. The same holds for regular closed sets. In contrast, the set of all uncountable closed subsets of $T$ is a non-Borel subset of $\F(T)$, unless $T$ is countable.

The sets $\{F_1\in\F(T):F_1\subset F\}$ and $\{F_1\in\F(T):F_1\supset F\}$ (for a given closed $F\subset T$) are Borel subsets of $\F(T)$. Moreover, the set of all pairs $(F_1,F_2)\in\F(T)\times\F(T)$ satisfying the relation $F_1\subset F_2$ is Borel. In contrast, the relation $F_1\cap F_2=\emptyset$ leads generally to a non-Borel set of pairs. Moreover, the set $\{F_1\in\F(T):F_1\cap F=\emptyset\}$ (for a given closed $F$) is generally a non-Borel subset of $\F(T)$. For example, it happens if $T$ is the unit ball of an infinite-dimensional separable Hilbert space and $F$ is the unit sphere. Thus, the intersection operation treated as a map $(F_1,F_2)\mapsto F_1\cap F_2$ from $\F(T)\times\F(T)$ to $\F(T)$ is generally non-Borel. However, the union operation $(F_1,F_2)\mapsto F_1\cup F_2$ is Borel. And if $T$ is σ-compact (in particular, if $T=\R^n$) then the intersection operation is Borel.

(See [K, Sect. 12.C, 27.B].)

Criticism

A quote from [Dur, Sect. 1.4(c), p. 33]:

$(S,\mathcal S)$ is said to be nice if there is a 1-1 map $\varphi$ from $S$ into $\R$ so that $\varphi$ and $\varphi^{-1}$ are both measurable.

Such spaces are often called standard Borel spaces, but we already have too many things named after Borel. The next result shows that most spaces arising in applications are nice.

(4.12) Theorem. If $S$ is a Borel subset of a complete separable metric space $M$, and $\mathcal S$ is the collection of Borel subsets of $S$, then $(S,\mathcal S)$ is nice.

It is not specified in the definition, whether $\varphi(S)$ must be a Borel set, or not. The proof of the theorem provides just a Borel 1-1 map $\varphi:S\to\R$ without addressing measurability of the function $\varphi^{-1}$ and the set $\varphi(S)$. (A complete proof would be considerably harder.) Later, in the proof of Theorem (1.6) of [Dur, Sect. 4.1(c)], measurability of $\varphi^{-1}$ and $\varphi(S)$ is used (see the last line of the proof).

References

[K]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597  Zbl 0819.04002
[D]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
[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
[Dur]	Richard Durrett, "Probability: theory and examples", second edition, Duxbury Press (1996).   MR1609153