A simple Efimov space with sequentially-nice space of probability measures

TL;DR

This paper constructs a special topological space under set-theoretic assumptions where the space of nonatomic probability measures exhibits desirable sequential and compactness properties, impacting the structure of continuous functions.

Contribution

It introduces a simple Efimov space with a sequentially-nice space of probability measures, linking set theory, topology, and functional analysis in a novel way.

Findings

The space of nonatomic probability measures is first-countable and sequentially compact.

The space of all probability measures is selectively sequentially pseudocompact.

The Banach space of continuous functions has the Gelfand-Phillips property.

Abstract

Under Jensen's diamond principle $\diamondsuit$, we construct a simple Efimov space $K$ whose space of nonatomic probability measures $P_{na}(K)$ is first-countable and sequentially compact. These two properties of $P_{na}(K)$ imply that the space of probability measures $P(K)$ on $K$ is selectively sequentially pseudocompact and the Banach space $C(K)$ of continuous functions on $K$ has the Gelfand-Phillips property. We show also that any sequence of probability measures on $K$ that converges to an atomic measure converges in norm, and any sequence of probability measures on $K$ converging to zero in sup-norm has a subsequence converging to a nonatomic probability measure.