Quasi-Measurable Spaces: A Convenient Foundation of Probability Theory

Abstract

We introduce the categories of quasi-measurable spaces, mild generalizations of quasi-Borel spaces that allow for general sample spaces and less restrictive random variables. Each such category is complete, cocomplete, cartesian closed and, as we show, even locally cartesian closed, a quasitopos and a Heyting category, and so models an extensional dependent type theory with a typed intuitionistic first-order logic. We put one probability monad in the foreground, the push-forward probability monad $\mathcal{P}$, and isolate the two properties of the sample space its theory needs: a Fubini property, making $\mathcal{P}$ commutative, and a product isomorphism $\Omega \times \Omega \cong \Omega$, making it strong. It is affine always. Our main interest is the category of quasi-universal spaces, where the sample space is the universal Hilbert cube $[0,1]^\mathbb{N}$ with the $\sigma$-algebra of all universally measurable subsets. That completion is what buys explicit descriptions of the induced $\sigma$-algebras, as intersections of Lebesgue-complete $\sigma$-algebras, and for countably separated spaces as the completion of any countable separating family, which quasi-Borel spaces do not admit. All push-forward probability monads collapse into one here, and on standard quasi-universal spaces the Giry monad joins them. We prove a Fubini theorem, a disintegration theorem for Markov kernels, Kolmogorov extension theorems and a conditional de Finetti theorem, and translate them into properties of the associated Markov category. Finally we formalize causal Bayesian networks over quasi-universal spaces and prove a global Markov property, translating transitional conditional independence into this setting and establishing its asymmetric separoid rules. Exponential objects then put variables and causal mechanisms on the same graphical footing, in partially generic causal Bayesian networks.