View source for Standard Borel space

''Also: standard measurable space''

{{MSC|03E15|28A05,54H05}}

[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]

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A [[measurable space|Borel space]] $(X,\A)$ is called '''standard''' if it satisfies the following equivalent conditions:
* $(X,\A)$ is [[Measurable space#isomorphic|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 [[Measurable space#product|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 [[Measurable space#separated|countably separated]], therefore not standard.)

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

The [[Measurable space#disjoint union|disjoint union]] of two standard Borel spaces is a standard Borel space.

(See {{Cite|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 {{Cite|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 {{Cite|M|Th. 3.2}} or {{Cite|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 [[Measurable space#separating|separating]] subset of $\A$. (See {{Cite|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 [[A-set|analytic set]], and it is [[Perfect measure|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 {{Cite|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 {{Cite|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 [[Measure space#Completion|completion]] of $(X,\A,\mu)$ is a [[standard probability space]]. (See {{Cite|K|Sect. 17.F}}.)

Let $(X,\A)$ be a standard Borel space, $\Mu$ the set of all probability measures on $(X,\A)$, and $\B$ the σ-algebra on $\Mu$ 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 $(\Mu,\B)$ 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 {{Cite|K|Sect. 17.E}}.)

====Space of closed sets====

Let $T$ be a complete separable metric space, and $\F(T)$ the [[Measurable space#Space of closed sets|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 {{Cite|K|Sect. 12.C, 27.B}}.)

====Criticism====

A quote from {{Cite|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 {{Cite|Dur|Sect. 4.1(c)}},  measurability of $\varphi^{-1}$ and $\varphi(S)$ is used (see the last  line of the proof).

====References====

{|
|valign="top"|{{Ref|K}}|| Alexander S.  Kechris, "Classical  descriptive set theory", Springer-Verlag (1995). &nbsp;  {{MR|1321597}} &nbsp;{{ZBL|0819.04002}} 
|-
|valign="top"|{{Ref|D}}|| Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989). &nbsp; {{MR|0982264}} &nbsp; {{ZBL|0686.60001}}
|-
|valign="top"|{{Ref|M}}||  George W. Mackey,  "Borel structure in groups and their duals",  ''Trans.  Amer. Math. Soc.''  '''85''' (1957), 134–165. &nbsp;  {{MR|0089999}} &nbsp;   {{ZBL|0082.11201}}
|-
|valign="top"|{{Ref|S}}||  S.M. Srivastava, "A course on Borel sets", Springer-Verlag (1998).  &nbsp; {{MR|1619545}} &nbsp;{{ZBL|0903.28001}}
|-
|valign="top"|{{Ref|Dur}}|| Richard Durrett, "Probability: theory and  examples", second edition, Duxbury Press (1996).  &nbsp;  {{MR|1609153}}
|}