# nLab quasi-Borel space

Quasi-Borel Space

### Context

#### Measure and probability theory

measure theory

probability theory

# 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 $M_X$.

## 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.)

Quasi-Borel spaces were introduced in