A Topologically Enriched Probability Monad on the Cartesian Closed Category of CGWH Spaces

Abstract

Probability monads on categories of topological spaces are classical objects of study in the categorical approach to probability theory, with important applications in the semantics of probabilistic programming languages. We construct a probability monad on the category of compactly generated weakly Hausdorff (CGWH) spaces, a (if not the) standard choice of convenient category of topological spaces. Because a general version of the Riesz representation theorem adapted to this setting plays a fundamental role in our construction, we name it the Riesz probability monad. We show that the Riesz probability monad is a simultaneous extension of the classical Radon and Giry monads that is topologically enriched. Topological enrichment corresponds to a strengthened continuous mapping theorem (in the sense of probability theory). In addition, restricting the Riesz probability monad to the Cartesian closed subcategory of weakly Hausdorff quotients of countably based (QCB) spaces results in a probability monad which is strongly affine, ensuring that the notions of independence and determinism interact as we would expect.