Measurable space

Also: Borel space

2020 Mathematics Subject Classification: Primary: 28A05 Secondary: 03E1554H05 [MSN][ZBL]

$ \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\P}{\mathbf P} $ 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.

Basic notions

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 $p_i:X\to X_i$ [3, 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}) $ [1, 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 $B\in\A_1$).

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. Every measurable set is saturated (that is, $x\sim y$ implies $x\in A \Longleftrightarrow y\in A$). If the set of equivalence classes is finite or countable 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 $\A$.

If $(X,\A)$ is countably generated then the cardinality of $\A$ is at most continuum [1, 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 $(X,\A)$ may be called

• separated if $\{A\in\A:x\in A\}=\{A\in\A:y\in A\}$ 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 $x,y\in X$.

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

Relations to topological spaces

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.

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.

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 [3, Sect. 9.C], [4, 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 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 $x,y\in A_n$).

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 [3, 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 [5, Sect.7.1], [6, Sect.51]. For more general (in particular, uncountable discrete) topological spaces the definitions of [5] and [6] disagree. Note also the σ-algebra of universally measurable sets.

Older terminology

Weaker assumptions on $\A$ were usual in the past. For example, according to [6], $\A$ need not contain the whole $X$, it is a σ-ring, not necessarily a σ-algebra. According to [7], 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).

References

[1]	Terence Tao, "An introduction to measure theory", AMS (2011) | MR2827917 | Zbl 05952932
[2]	David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002) | MR1873379 | Zbl 0992.60001
[3]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995) | MR1321597 | Zbl 0819.04002
[4]	Howard Becker and Alexander S. Kechris, "The descriptive set theory of Polish group actions", Cambridge (1996) | MR1425877 | Zbl 0949.54052
[5]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) | MR0982264 | Zbl 0686.60001
[6]	Paul R. Halmos, "Measure theory", v. Nostrand (1950) | MR0033869 | Zbl 0040.16802
[7]	Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) | MR0055409 | Zbl 0052.05301