Monolithic spaces of measures

TL;DR

This paper investigates the topological properties of the space of probability measures on a compact space, focusing on conditions under which this measure space is $ ext{aleph}_0$-monolithic, revealing complex interactions between measure and topology.

Contribution

It characterizes when the space of probability measures on a compact space is $ ext{aleph}_0$-monolithic, including a consistency result involving $ ext{diamondsuit}$ and Corson compact spaces.

Findings

Existence of a nonseparable Corson compact space with $ ext{aleph}_0$-monolithic measure space.

Construction under $ ext{diamondsuit}$ of such a space supporting an uncountable measure.

Insights into the relationship between measure support and topological properties of $K$.

Abstract

For a compact space $K$ we consider the space $P(K)$, of probability regular Borel measures on $K$, equipped with the $weak^\ast$ topology inherited from $C(K)^\ast$. We discuss possible characterizations of those compact spaces $K$ for which $P(K)$ is $\aleph_0$-monolithic. The main result states that under $\diamondsuit$ there exists a nonseparable Corson compact space $K$ such that $P(K)$ is $\aleph_0$-monolithic but $K$ supports a measure of uncountable type.