quasi-Borel space

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.


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