[Paper Review] Topology of probability measure spaces, II

Тарас Банах|arXiv (Cornell University)|Jun 8, 2012

Advanced Topology and Set Theory2 references3 citations

TL;DR

This paper investigates the barycenter map for Radon probability measures on locally convex spaces, proving its continuity if and only if the underlying set is bounded. It establishes that the functor $ς$ of Radon probability measures is monadic over the category of metrizable spaces and extends the functors $P_{\tau}$ and $\hat{P}$ to the categories of bounded metric spaces and uniform spaces, analyzing their metrizability and uniform continuity properties.

ABSTRACT

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 $P_τ$ of spaces of probability $τ$-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 $P_τ$ 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.

Motivation & Objective

• To investigate the barycenter map for spaces of Radon probability measures and its continuity properties.
• To prove that the functor $\hat{P}$ is monadic in the category of metrizable spaces using the barycenter map.
• To extend the functors $P_{\tau}$ and $\hat{P}$ to the category of bounded metric spaces ($\mathcal{BM}etr$) and uniform spaces ($\mathcal{U}nif$).
• To analyze the metrizability and uniform continuity of the lifted functors in $\mathcal{BM}etr$ and $\mathcal{U}nif$.
• To address open questions on preservation of completeness-type properties by $P_{\tau}$ and $\hat{P}$, particularly in relation to Dieudonné and Hewitt completeness.

Proposed method

• Define the barycenter map $b_X: \hat{P}(X) \to E^{**}$ for a weakly bounded subset $X$ of a locally convex space $E$, where $b_X(\mu)(f) = \mu(f|X)$ for $f \in E^*$.
• Prove that $b_X$ is affine and satisfies $b_X(\delta_x) = x$ for all $x \in X$.
• Establish that $b_X$ is continuous if and only if $X$ is bounded in $E$, using topological arguments involving neighborhoods and weakly bounded sets.
• Lift the functors $P_{\tau}$ and $\hat{P}$ to the category $\mathcal{BM}etr$ via bounded pseudometrics $p_\tau$, inducing uniform structures $\mathcal{U}_\tau$.
• Show that the map $P_{\tau}(f)$ is uniformly continuous for any uniformly continuous map $f$ between bounded metric spaces.
• Use uniform embeddings and the Ascoli theorem to prove that $P_{\tau}(X)$ is totally bounded if and only if $X$ is totally bounded in the uniform space setting.

Experimental results

Research questions

• RQ1 Under what conditions is the barycenter map $b_X: \hat{P}(X) \to E^{**}$ continuous?
• RQ2 Is the functor $\hat{P}$ monadic in the category of metrizable spaces, and how does the barycenter map support this?
• RQ3 Can the functors $P_{\tau}$ and $\hat{P}$ be lifted to the category of bounded metric spaces and uniform spaces, and what properties do these liftings preserve?
• RQ4 Does the functor $\hat{P}$ preserve completeness in uniform spaces, and what are the limitations in this context?
• RQ5 What is the relationship between total boundedness of a uniform space $X$ and total boundedness of $P_{\tau}(X)$?

Key findings

• The barycenter map $b_X: \hat{P}(X) \to E^{**}$ is continuous if and only if the set $X \subset E$ is bounded.
• The functor $\hat{P}$ is monadic over the category of metrizable spaces, as established via the barycenter map and its properties.
• The functors $P_{\tau}$ and $\hat{P}$ admit liftings to the category $\mathcal{BM}etr$ of bounded metric spaces, preserving uniform continuity of maps.
• The functors $P_{\tau}$ and $\hat{P}$ can also be lifted to the category $\mathcal{U}nif$ of uniform spaces, with $P_{\tau}(f)$ being uniformly continuous for any uniformly continuous $f$.
• A uniform space $(X, \mathcal{U})$ is totally bounded if and only if $(P_{\tau}(X), \mathcal{U}_\tau)$ is totally bounded.
• The functor $\hat{P}$ does not generally preserve complete uniform spaces, as demonstrated by a Cauchy net in $\hat{P}(\mathbb{R}^A)$ without an accumulation point for uncountable $A$.