Quasi-Measurable Spaces

Abstract

We introduce the categories of quasi-measurable spaces, which are slight generalizations of the category of quasi-Borel spaces, where we now allow for general sample spaces and less restrictive random variables, quasi-measurable spaces and maps. We show that each category of quasi-measurable spaces is bi-complete and cartesian closed. We also introduce several different strong probability monads. Together these constructions provide convenient categories for higher probability theory that also support semantics of higherorder probabilistic programming languages in the same way as the category of quasi-Borel spaces does.An important special case is the category of quasi-universal spaces, where the sample space is the set of the real numbers R together with the σ-algebra of all universally measurable subsets. The induced σ-algebras on those quasiuniversal spaces then have explicit descriptions in terms of intersections of Lebesgue-complete σ-algebras. A central role is then played by countably separated quasi-universal spaces and universal quasi-universal spaces, which replace the role of standard Borel measurable spaces. We then go on and prove in this setting, besides others, a Fubini theorem, a disintegration theorem for Markov kernels, a Kolmogorov extension theorem and a conditional de Finetti theorem. We also translate our findings into properties of the corresponding Markov category of Markov kernels between universal quasiuniversal spaces.Furthermore, we formalize probabilistic graphical models like causal Bayesian networks in terms of quasi-universal spaces and prove a global Markov property for them. For this we need to translate the notion of transitional conditional independence into this setting and study its (asymmetric) separoid