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 |
Standard Borel space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_Borel_space&oldid=43097