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

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.