Topology of probability measure spaces, II

This paper is a follow-up to the author’s work “Topology of probability measure space, I” devoted to investigation of the functors $\hat P$ and $P_\tau$ of spaces of probability $\tau$-smooth and Radon measures. In this part, we study the barycenter map for spaces of Radon probability measures. The obtained results are applied to show that the functor $\hat P$ is monadic in the category of metrizable spaces. Also we show that the functors $\hat P$ and $P_\tau$ admit liftings to the category $BMetr$ of bounded metric spaces and also to the category $Unif$ of uniform spaces, and investigate properties of those liftings.

💡 Research Summary

The paper continues the investigation of the functors \hat P and P_τ initiated in “Topology of probability measure space, I”. Its central focus is the barycenter (or centre) map for spaces of Radon probability measures and the categorical consequences that follow. After recalling the definitions of Radon measures and τ‑smooth measures, the author establishes that for any completely regular (in particular, metrizable) space X, the barycenter map b_X : \hat P(X) → X, which sends a Radon probability measure to its expectation (or centre of mass), is well‑defined, affine, and continuous. The proof relies on the regularity of Radon measures, the Hahn–Banach separation theorem, and the fact that the space of continuous bounded functions separates points in X. When X is a complete metric space, b_X is shown to be a retraction onto the closed convex hull of X, and it is unique among affine continuous maps.

Using the barycenter map, the author constructs the monad structure of the functor \hat P on the category Metr of metrizable spaces. The unit η_X : X → \hat P(X) is the Dirac embedding x ↦ δ_x, and the multiplication μ_X : \hat P(\hat P(X)) → \hat P(X) is defined by integrating a measure of measures: for ν ∈ \hat P(\hat P(X)), μ_X(ν) = ∫_{ \hat P(X)} m dν(m). The paper verifies the monad axioms (unit laws and associativity) and proves that both η_X and μ_X are continuous with respect to the topology of weak convergence on Radon measures. The continuity of μ_X follows from the continuity of the barycenter map and the fact that integration against a Radon measure preserves weak convergence.

The second major contribution is the lifting of the functors \hat P and P_τ to the categories BMetr (of bounded metric spaces) and Unif (of uniform spaces). For a bounded metric space (X,d), a metric \hat d on \hat P(X) is introduced, which coincides with the Kantorovich–Rubinstein (Wasserstein‑1) distance: \