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, 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 higher-order 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 together with the sigma-algebra of all universally measurable subsets. The induced sigma-algebras on those quasi-universal spaces then have explicit descriptions in terms of intersections of Lebesgue-complete sigma-algebras. A central role is then played by countably separated and universal quasi-universal spaces, which replace the role of standard Borel spaces. We prove in this setting 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 quasi-universal spaces. Furthermore, we formalize causal Bayesian networks in terms of quasi-universal spaces and prove a global Markov property. For this we translate the notion of transitional conditional independence into this setting and study its (asymmetric) separoid rules. Altogether we are now able to reason about conditional independence relations between variables and causal mechanisms on equal footing. Finally, we also highlight how one can use exponential objects and random functions for counterfactual reasoning.