Complete Quasi-Metrics for Hyperspaces, Continuous Valuations, and Previsions

Abstract

The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in that case it metrizes the weak topology. We introduce a variant of that construction in the realm of quasi-metric spaces, and prove that it is algebraic Yoneda-complete as soon as the underlying quasi-metric space of points is algebraic Yoneda-complete, and that the associated topology is the weak topology. We do this not only for probability distributions, represented as normalized continuous valuations, but also for subprobability distributions, for various hyperspaces, and in general for different brands of functionals. Those functionals model probabilistic choice, angelic and demonic non-deterministic choice, and their combinations. The mathematics needed for those results are more demanding than in the simpler case of metric spaces. To obtain our results, we prove a few other results that have independent interest, notably: continuous Yoneda-complete spaces are consonant; on a continuous Yoneda-complete space, the Scott topology on the space of $\overline{\mathbb{R}}_+$-valued lower semicontinuous maps coincides with the compact-open and Isbell topologies, and the subspace topology on spaces of $\alpha$-Lipschitz continuous maps also coincides with the topology of pointwise convergence, and is stably compact; we introduce and study the so-called Lipschitz-regular quasi-metric spaces, and we show that the formal ball functor induces a Kock-Z\"oberlein monad, of which all algebras are Lipschitz-regular; and we prove a minimax theorem where one of the spaces is not compact Hausdorff, but merely compact.