nLab quasi-Borel space




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.


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

  1. M XM_X contains all constant functions.
  2. M XM_X is closed under composition with measurable functions: if f:f : \mathbb{R} \to \mathbb{R} is measurable and αM X\alpha \in M_X, αfM X\alpha \circ f \in M_X.
  3. M XM_X is closed under gluing functions with disjoint Borel domains: for any partition = iS i\mathbb{R} = \biguplus_{i\in\mathbb{N}} S_i by Borel S iS_i, and {α iM X} i\{\alpha_i \in M_X\}_{i\in\mathbb{N}}, then the function β(x)=α i(x)\beta(x) = \alpha_i(x) when xS ix \in S_i is in M XM_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 22 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.


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).

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

The right adjoint takes a measurable space XX to the quasi-Borel space where M XM_X comprises the measurable functions. The left adjoint regards a quasi-Borel space XX with the sigma-algebra comprising all those sets UXU\subseteq X for which the characteristic function X2X\to 2 is a morphism.

Probability distributions

A probability measure on a quasi-Borel space is defined to be a function X\mathbb{R}\to X in M XM_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][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.


Quasi-Borel spaces were introduced in

Last revised on January 25, 2024 at 15:22:13. See the history of this page for a list of all contributions to it.