# Measurable space

Also: Borel space

2010 Mathematics Subject Classification: Primary: 28A05 Secondary: 03E1554H05 [MSN][ZBL]

$\newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M}$ A measurable space is a set with a distinguished σ-algebra of subsets (called measurable). More formally, it is a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.

Examples: $\R^n$ with the Borel σ-algebra; $\R^n$ with the Lebesgue σ-algebra.

Warning. In contrast to measure spaces, in this context (a) no measure is given; (b) no subset is called negligible (null); (c) measurability of a subset $A\subset X$ means just $A\in\A$.

#### Basic notions and constructions

Let $(X,\A)$ and $(Y,\B)$ be measurable spaces.

• A map $f:X\to Y$ is called measurable if $f^{-1}(B) \in \A$ for every $B\in\B$.
• These two measurable spaces are called isomorphic if there exists a bijection $f:X\to Y$ such that $f$ and $f^{-1}$ are measurable (such $f$ is called an isomorphism).

Let $X$ be a set, $(Y,\B)$ a measurable space, and $(f_i)_{i\in I}$ a family of maps $f_i:X\to Y$. The σ-algebra generated by these maps is defined as the smallest σ-algebra $\A$ on $X$ such that all $f_i$ are measurable from $(X,\A)$ to $(Y,\B)$. More generally, one may take measurable spaces $(Y_i,\B_i)$ and maps $f_i:X\to Y_i$. On the other hand, if $Y$ is $\R$ (or $\C$, $\R^n$ etc.) then $\B$ is by default the Borel σ-algebra.

Given a family of measurable spaces $(X_i,\A_i)$ for $i\in I$, their product is defined as the measurable space $(X,\A)$ where $X=\prod_i X_i$ is the direct product of sets, and $\A$ is generated by the projection maps $p_i:X\to X_i$ [K, Sect. 10.B].

Example and warning. Denoting the Borel σ-algebra on $\R^n$ by $\B_n$ and the Lebesgue σ-algebra on $\R^n$ by $\A_n$ we have $(\R^m,\B_m) \times (\R^n,\B_n) = (\R^{m+n},\B_{m+n})$ but $(\R^m,\A_m) \times (\R^n,\A_n) \ne (\R^{m+n},\A_{m+n})$ [T, Exercise 1.7.19]. Denoting $(\R^m,\A_m) \times (\R^n,\A_n) = (\R^{m+n},\A_{m,n})$ we have for every $A \in \A_{m,n}$ $\{y\in\R^n:(x,y)\in A\} \in \B_n \quad \text{for almost all } x\in\R^m.$ In particular, a set of the form $\{(x,y)\in\R^2:x-y\in B\}$ belongs to $\A_{1,1}$ if and only if $B\in\B_1$ (rather than $B\in\A_1$).

Example and warning. Given a one-to-one map $f:\R\to\R$, we consider the indicator of its graph: $g(x,y)=1$ when $y=f(x)$, otherwise $g(x,y)=0$. The function $y\mapsto g(x,y)$ is Borel measurable for every $x$ (being just the indicator of a single point), and the function $x\mapsto g(x,y)$ is Borel measurable for every $y$. In other words, $g(x,y)$ is Borel measurable in $x$ and $y$ separately. Nevertheless $g$ is not Borel measurable, unless $f$ is. In other words, $g(x,y)$ need not be jointly measurable in $x$ and $y$.

Example and warning. The set $\R^X$ of all functions $X\to\R$ may be thought of as the product of copies of $\R$; the corresponding σ-algebra on $\R^X$ is generated by the evaluation maps $f\mapsto f(x)$ for all $x\in X$. However, for uncountable $X$ this approach is less useful than it may seem, because $f(x)$ fails to be jointly measurable in $f$ and $x$ even if all subsets of $X$ are measurable. That is, the map $(f,x)\mapsto f(x)$ from $\R^X\times X$ to $\R$ is not measurable.

Let $(X,\A)$ be a measurable space and $Y\subset X$ a subset (not necessarily measurable). Introducing $\B=\{A\cap Y:A\in\A\}$ one gets a measurable space $(Y,\B)$ called a (measurable) subspace of $(X,\A)$.

Let $(X_1,\A_1)$ and $(X_2,\A_2)$ be measurable spaces, $X_1\cap X_2=\emptyset$. Introducing $X=X_1\cup X_2$ and $\A=\{A_1\cup A_2:A_1\in\A_1,A_2\in\A_2\}$ one gets a measurable space $(X,\A)$ called the (disjoint) union of $(X_1,\A_1)$ and $(X_2,\A_2)$.

Let $(X,\A)$ be a measurable space and $r$ an equivalence relation on $X$. Denoting by $Y$ the set of all equivalence classes and introducing $\B=\{B\subset Y:p^{-1}(B)\in\A\}$ where $p:X\to Y$ is the projection, one gets a measurable space $(Y,\B)$ called the quotient (measurable) space of $(X,\A)$ (by $r$).

Given a measurable space $(X,\A)$, an equivalence relation $\stackrel{\A}{\sim}$ on $X$, defined by $x\stackrel{\A}{\sim}y \quad \text{means} \quad \forall A\in\A \; (\,x\in A \Longleftrightarrow y\in A\,),$ leads to a partition of $X$ into equivalence classes, so-called atoms of $(X,\A)$ (not always measurable, see [C, Sect. 8.6]). Every measurable set is saturated (that is, $x\sim y$ implies $x\in A \Longleftrightarrow y\in A$). If the atoms are a finite or countable set then all saturated sets are measurable. But in general saturated sets are more than a σ-algebra; an arbitrary (not just countable) union of saturated sets is a saturated set.

#### Some classes of measurable spaces

A measurable space $(X,\A)$ (as well as its σ-algebra $\A$) is called countably generated if $\A$ is generated by some countable subset of $\A$.

The product of a finite or countable family of countably generated measurable spaces is countably generated.

If $(X,\A)$ is countably generated then the cardinality of $\A$ is at most continuum [T, Exercise 1.4.16].

Example: $\R^n$ with the Borel σ-algebra is countably generated; $\R^n$ with the Lebesgue σ-algebra is not. Every countably generated sub-σ-algebra $\A_0$ of the Lebesgue σ-algebra is almost Borel in the following sense: there exists a Borel set $B_0$ of full measure such that $A\cap B_0$ is a Borel set for every $A\in\A_0$. The Borel σ-algebra is of cardinality continuum; the Lebesgue σ-algebra is of higher cardinality (since it contains all subset of a null set of cardinality continuum).

A measurable space $(X,\A)$ is called

• separated (in other words, separating points) if the corresponding equivalence relation is the equality, that is, $\{A\in\A:x\in A\}=\{A\in\A:y\in A\}$ implies $x=y$ for $x,y\in X$;
• countably separated if there exists a sequence of sets $A_n\in\A$ such that $\{n:x\in A_n\}=\{n:y\in A_n\}$ implies $x=y$ for $x,y\in X$ (so-called separating sequence).

(See [C, Sect. 8.6].)

If $(X,\A)$ is separated and $X$ is finite or countable then all subsets of $X$ are measurable.

Example: $\R^n$ with the Borel σ-algebra is countably separated; the same holds for the Lebesgue σ-algebra.

Let $(X,\A)$, $(Y,\B)$ be measurable spaces, $f:X\to Y$ a measurable map, and $(Y,\B)$ countably separated. Then the graph $\{(x,f(x)):x\in X\}$ of $f$ is a measurable subset of $X\times Y$. (See [K, Sect. 12.A].)

A much deeper theory is available for standard, analytic and universally measurable measurable spaces (see the separate articles).

#### Relations to measures and integrals

An integral in one variable is measurable in the other variable(s) in the following sense.

Let $(X,\A)$ and $(Y,\B)$ be measurable spaces, $\mu$ a finite measure on $(Y,\B)$, and $f:X\times Y\to\R$ a bounded measurable function. Then the function $g:X\to\R$ defined by $g(x) = \int f(x,y) \, \mu(\!\rd y)$ is measurable.

Moreover, the integral is jointly measurable in $x$ and $\mu$ in the following sense.

The formula $G(x,\mu) = \int f(x,y) \, \mu(\!\rd y)$ defines a measurable function $G:X\times\M(Y)\to\R$, where $\M(Y)$ is the set of all finite measures on $Y$, endowed with the σ-algebra generated by the maps $\mu\mapsto\mu(B)$ for all $B\in\B$. (See [K, Sect. 17.E].)

Thus, the measure $\mu$ may be treated as another variable. Also the function $f$ may be treated as a variable provided, however, that $f(y)$ is jointly measurable in $f$ and $y$, which fails in general but holds for continuous functions, see below.

#### Relations to topological spaces and continuity

##### Borel sets

Every topology generates a σ-algebra, called Borel σ-algebra. That is, the Borel σ-algebra on a topological space is, by definition, generated by the open sets. This σ-algebra is used, unless the contrary is explicitly stated. Accordingly, one says "Borel measurable" or just "Borel" instead of "measurable" (sets and maps).

Example. The following three σ-algebras on a separable Hilbert space $H$ are equal:

• the σ-algebra generated by the linear functionals $x \mapsto \langle x,y \rangle$ for $y\in H$;
• the Borel σ-algebra corresponding to the norm topology on $H$;
• the Borel σ-algebra corresponding to the weak topology on $H$.

That is instructive: topological spaces are not a prerequisite to measurable spaces.

##### Joint measurability

Let $X$ be a measurable space, $Y$ a separable metric space (or just a second countable topological space) and $f:X\times Y\to\R$ a function such that $f(x,y)$ is measurable in $x$ and continuous in $y$; then $f(x,y)$ is jointly measurable in $x$ and $y$. (See [K, Sect. 11.C].)

Taking $X=C(Y)$, the space of all continuous functions $Y\to\R$, endowed with the σ-algebra generated by the evaluation maps $f\mapsto f(y)$ for all $y\in Y$, we conclude that $f(y)$ is jointly measurable in $f$ and $y$.

The joint measurability of $\int F(x,y)\,\mu(\!\rd y)$ in $x$ and $\mu$ (see above), applied to $X=C(Y)$ and $F(f,y)=f(y)$, gives the joint measurability of $\int f(y)\,\mu(\!\rd y)$ in $f$ and $\mu$ (as long as $f$ is continuous on a second countable topological space).

##### Semicontinuity

A real-valued function $f$ on a topological space $T$ is called upper semicontinuous if the set $\{t\in T:f(t)<a\}$ is open for all $a\in\R$. Such functions are Borel measurable.

In particular, the indicator $\bsone_A$ of a set $A\subset T$ (equal $1$ on $A$ and $0$ on $T\setminus A$) is upper semicontinuous if and only if $A$ is closed.

The set $C^\text{upper}(T)$ of all upper semicontinuous functions $T\to\R$ becomes a measurable space, being endowed with the σ-algebra generated by the maps $f\mapsto\sup_U f$ for all open $U\subset T$.

If $T$ is a separable metric space (or just a second countable topological space) then $f(t)$ is jointly measurable in $f\in C^\text{upper}(T)$ and $t\in T$. (Hint: $f(t)=\inf_{n:t\in U_n} \sup_{U_n} f$ where $(U_n)_n$ is a countable base on $T$.)

Thus, the joint measurability of $\int f\rd\mu$ in $f$ and $\mu$ holds also for $f\in C^\text{upper}(T)$.

Warning. The evaluation maps $f\mapsto f(t)$ for $t\in T$ generate a smaller σ-algebra on $C^\text{upper}(T)$; this smaller σ-algebra is less useful, since it fails to make $f(t)$ jointly measurable. (Hint: its restriction to indicators of single-point sets contains only countable sets and their complements.)

##### Space of closed sets

The set $\F(T)$ of all closed subsets of a topological space $T$ is embedded into $C^\text{upper}(T)$ by indicators, $\F(T)\ni F\mapsto\bsone_F \in C^\text{upper}(T)$. Thus, $\F(T)$ inherits from $C^\text{upper}(T)$ a σ-algebra and becomes a measurable space, so-called Effros Borel space (over $T$). Its σ-algebra is generated by sets $\{F\in\F(T):F\cap U\neq\emptyset\}$ for all open sets $U\subset T$. The set $\{(F,t):t\in F\}$ is a measurable subset of $\F(T)\times T$ provided that $T$ is second countable. (See [K, Sect. 12.C].)

##### Topological groups

A Borel measurable map is generally not continuous, and a Borel isomorphism is generally not a homeomorphism. However, every Borel measurable homomorphism between Polish groups is continuous. Accordingly, the topology of a Polish group is uniquely determined by its Borel σ-algebra (see [K, Sect. 9.C], [BK, Sect. 1.2]).

Example. Consider again a separable Hilbert space $H$.

• Every Borel measurable linear functional $H\to\C$ is continuous.
• Every Borel measurable linear operator $H\to H$ is continuous.
• A set $U\subset H$ contains a neighborhood of the origin (in the norm topology) if and only if there exists a sequence of Borel sets $A_1,A_2,\dots\subset H$ such that $A_1\cup A_2\cup\dots=H$ and $A_n-A_n\subset U$ for all $n$ (that is, $x-y\in U$ for all $x,y\in A_n$).
##### Some other σ-algebras

The Borel σ-algebra is not the only bridge between topological and measurable spaces. All sets having the Baire property (sometimes called Baire sets, which may be confusing) are a σ-algebra (generated by open sets together with meager sets) greater than Borel [K, Sect.8.F]. On the other hand, all compact $G_\delta$ subsets of a compact Hausdorff topological space generate a σ-algebra (smaller than Borel) of sets called Baire sets in [D, Sect.7.1], [H, Sect.51]. For more general (in particular, uncountable discrete) topological spaces the definitions of [D] and [H] disagree. Note also the σ-algebra of universally measurable sets.

#### On terminology

"Borel space" and "measurable space" are often used as synonyms. But according to [K, Sect. 12.A] a Borel space is a countably generated measurable space that separates points (or equivalently, a measurable space isomorphic to a separable metric space with the Borel σ-algebra), in which case "Borel" instead of "measurable" applies also to sets and maps.

Weaker assumptions on $\A$ were usual in the past. For example, according to [H], $\A$ need not contain the whole $X$, it is a σ-ring, not necessarily a σ-algebra. According to [R], a measurable space is not a pair $(X,\A)$ but a measure space $(X,\A,\mu)$ such that $X\in\A$ (and again, $\A$ is generally a σ-ring).