  Probability monads on categories of topological spaces are classical objects
of study in the categorical approach to probability theory. 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 novel version of the Riesz representation theorem
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.
