Stone-Cech extensions of probability measure spaces

TL;DR

This paper characterizes when the space of probability Radon measures on a topological space is pseudocompact, showing a key equivalence involving Stone-Cech extensions and providing conditions for pseudocompactness.

Contribution

It establishes a precise criterion for the pseudocompactness of measure spaces in relation to Stone-Cech extensions and constructs examples where this property fails.

Findings

$P(eta X)=eta P(X)$ iff $P(X)$ is pseudocompact

Constructs a locally compact pseudocompact space with non-pseudocompact measure space

Provides conditions under which $P(X)$ is pseudocompact

Abstract

It is proved that $P(\beta X)=\beta P(X)$ if and only if $P(X)$ is a pseudocompact space, where $P(X)$ is the space of probability Radon measures with weak topology and $\beta X$ is a Stone-Cech extension of the space $X$. A locally compact pseudocompact space $X$ is constructed such that $P(X)$ is not pseudocompact. Conditions are obtained under which $P(X)$ is pseudocompact.