nLab standard Borel space

Idea

Standard Borel spaces are particular measurable spaces, well suited for a number of constructions in traditional probability theory, such as for forming conditional probability.

From the point of view of categorical probability they form a particularly well-behaved Markov category, BorelStoch.

Definitions

A topological space is called completely separably metrizable if the topology can be metrized by a complete and separable metric. These spaces are also called Polish spaces, see there for more information.

A measurable space is called a standard Borel space if it can be written as a Polish space with its Borel sigma-algebra.

By analogy, a measure space (for example a probability space) is called a standard Borel measure space if and only if its underlying measurable space is standard Borel as above.

An alternative, but equivalent, definition of standard Borel spaces which is more useful for applications to probability theory (as opposed to analysis) requires some additional terminology.

Given a set $X$ the finite field (Boolean algebra) of $X$ generated by $n$ subsets of $X$, say $S_n$ for $n=0,1,...,n-1$, is denoted $\mathbb{F}_n = \{ S_0, S_1, ... , S_{n-1}\}$. The set of atoms of $\mathbb{F}_n$ are the set of all nonempty intersection sets of

$$\bigcap_{i=0}^{n-1} S_i^{\star}$$

where $S_i^{\star}$ is either $S_i$ or its complement $S_i^c$.

A standard Borel space $(X, \Sigma_X)$ consists of a set $X$ with a $\sigma$-algebra on $X$ generated by a field $\mathbb{F} = \cup_n \mathbb{F}_n$, so $\Sigma_X = \sigma(\mathbb{F})$, such that (1) $\mathbb{F}_n \subseteq \mathbb{F}_{n+1}$ for all $n$, and (2) If $\{A_n\}_{n}$ is a sequence of atoms such that $A_{n+1} \subseteq A_n$ for all $n$, then $\cap_{n} A_n \neq \emptyset$.

When such a sequence of finite fields $\{\mathbb{F}_n\}_n$ exists we say that the sequence of finite fields is a basis for the field $\mathbb{F}$, and the definition for a standard Borel space can be compactly stated as

A measurable space $(X, \Sigma_X)$ is a standard Borel space if and only if $\Sigma_X = \sigma(\mathbb{F})$ for some field $\mathbb{F}$ which possesses a basis.

Examples

• The real line is a standard Borel space.

• Every finite or countable set, equipped with the discrete sigma-algebra, is a standard Borel spaces.

Up to isomorphism of measurable spaces, these are the only examples. Note in particular that all finite powers of the real line and all intervals, as measurable spaces, are isomorphic to the real line.

Categories of standard Borel spaces

• The category BorelMeas has as objects standard Borel measurable spaces, and as morphisms measurable functions.

• The category BorelStoch has as objects standard Borel spaces, and as morphisms Markov kernels between them.

• The category of couplings has as objects standard Borel measure (not just measurable) spaces, and either equivalence classes of Markov kernels between them (under almost sure equality), or couplings between them.

Properties

• Finite and countable products (in the category Meas, i.e. with the tensor product sigma-algebra) of standard Borel spaces are again standard Borel.

• Measurable subsets of standard Borel spaces, with the induced sigma-algebra, are again standard Borel.

• Measurable retracts of a standard Borel space are again standard Borel.

• The sigma-algebra of a standard Borel space is countably generated and separating.

This has a number of consequences, for example:

• Every Markov kernel between standard Borel probability spaces admits a Bayesian inverse.

• Disintegration theorem: For every sub-sigma-algebra of a standard Borel space, conditional expectation gives rise to a regular conditional distribution.

• Standard Borel spaces admit countably infinite tensor products.

• The Giry monad preserves standard Borel spaces: if $X$ is a standard Borel space, its functorial image $P X$ is standard Borel too. Therefore the Giry monad restricts to a monad on BorelMeas, usually known under the same name.

Characterizations of BorelMeas

The Loomis–Sikorski duality theorem says that the category of standard Borel spaces is the opposite of the category of countably presented countably complete Boolean algebras.

Using this, Chen proved that the category of standard Borel spaces is the (bi)initial object in the 2-category of countably complete countably extensive Boolean categories:

• Ruiyuan Chen, A universal characterization of standard Borel spaces, Journal of Symbolic Logic 88(2) (2023), 510–539.

Related Concepts

• quasi-Borel space
• Polish space
• measurable space
• Stoch
• Markov kernel

References

The equivalence between the two definitions can be found in

• Chris Preston. Some notes on standard borel and related spaces arXiv:0809.3066

A gentle introduction to the alternative definition is given in

• Robert M. Gray. Probability, Random Processes, and Ergodic Properties. Springer-Verlag, 2008.

The proof that the Giry monad restricts to standard Borel spaces can be found in

• Kirk Sturtz, Deriving the Giry algebras on standard Borel spaces using $\mathbb{R}_{\infty}$-generalized points, $[$arXiv:2409.14861$]$

category: probability