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 consists of an underlying set and a set of functions satisfying:
A morphism of quasi-Borel spaces is a function that respects composition with these functions.
For example, is a quasi-Borel space, with the Borel functions. The two-element set 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?.
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 , equivalently, it is a measurable space that comes from a Polish space, equivalently, it is either isomorphic to or countable, discrete and non-empty.)
There is an adjunction between quasi-Borel spaces and measurable spaces (related to the nerve and realization construction).
The right adjoint takes a measurable space to the quasi-Borel space where comprises the measurable functions. The left adjoint regards a quasi-Borel space with the sigma-algebra comprising all those sets for which the characteristic function is a morphism.
A probability measure on a quasi-Borel space is defined to be a function in . 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 is regarded with the uniform distribution on .
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.