nLab quasi-Borel space

Idea

A quasi-Borel space (QBS) is a set equipped with a notion of random variable, providing a model of measurable space suitable for probability theory. The advantage of quasi-Borel spaces over traditional formulations is that they provide a nice category of measurable spaces: it is cartesian closed, and the set of probability measures of a QBS forms a QBS.

Definition

A quasi-Borel space $X$ consists of an underlying set $|X|$ and a set of functions $M_X \subseteq (\mathbb{R} \to |X|)$ satisfying:

1. $M_X$ contains all constant functions.
2. $M_X$ is closed under composition with measurable functions: if $f : \mathbb{R} \to \mathbb{R}$ is measurable and $\alpha \in M_X$, $\alpha \circ f \in M_X$.
3. $M_X$ is closed under gluing functions with disjoint Borel domains: for any partition $\mathbb{R} = \biguplus_{i\in\mathbb{N}} S_i$ by Borel $S_i$, and $\{\alpha_i \in M_X\}_{i\in\mathbb{N}}$, then the function $\beta(x) = \alpha_i(x)$ when $x \in S_i$ is in $M_X$.

A morphism of quasi-Borel spaces is a function that respects composition with these functions.

For example, $\mathbb{R}$ is a quasi-Borel space, with the Borel functions. The two-element set $2$ is a quasi-Borel space, with the functions that are characteristic functions of Borel subsets.

The category of quasi-Borel spaces is cartesian closed, unlike the category of measurable spaces.

Applications

The category of quasi-Borel spaces can be used as a denotational semantics for higher-order probabilistic programming languages?.

As a quasitopos of concrete sheaves

The category of quasi-Borel spaces is the category of concrete sheaves on the category of standard Borel spaces considered with the extensive coverage. As such, quasi-Borel spaces form a Grothendieck quasitopos. (A standard Borel space is a measurable space that is a retract of $\mathbb{R}$, equivalently, it is a measurable space that comes from a Polish space, equivalently, it is either isomorphic to $\mathbb{R}$ or countable, discrete and non-empty.)

Connection to measurable spaces

There is an adjunction between quasi-Borel spaces and measurable spaces (related to the nerve and realization construction).

$$Meas \stackrel{\leftarrow}{\rightarrow} Qbs,$$

The right adjoint takes a measurable space $X$ to the quasi-Borel space where $M_X$ comprises the measurable functions. The left adjoint regards a quasi-Borel space $X$ with the sigma-algebra comprising all those sets $U\subseteq X$ for which the characteristic function $X\to 2$ is a morphism.

Probability distributions

A probability measure on a quasi-Borel space is defined to be a function $\mathbb{R}\to X$ in $M_X$. This is regarded as a random variable. In particular we regard two probability measures as equal if the laws are the same probability distributions on the underlying measurable spaces, when $\mathbb{R}$ is regarded with the uniform distribution on $[0,1]$.

This leads to an affine, commutative monad on the category of quasi-Borel spaces. Restricted to standard Borel spaces, it agrees with the Giry monad.

The monad can be regarded as a probability monad, and its Kleisli category is a Markov category.

Related Concepts

• quasi-topological space
• subsequential space
• concrete sheaf
• standard Borel space

References

Quasi-Borel spaces were introduced in

• Chris Heunen, Ohad Kammar, Sam Staton and Hongseok Yang, A convenient category for higher-order probability theory, Logic in Computer Science 2017 (arXiv:1701.02547)