Sequential closure in the space of measures

TL;DR

This paper investigates the properties of measures on compact spaces, demonstrating the existence of measures with complex sequential closure properties and establishing conditions under which these closures are obtained in countably many steps.

Contribution

It introduces new examples of compact spaces with measures exhibiting high complexity in their sequential closure hierarchy and clarifies when these closures are countably attainable.

Findings

Existence of measures not approximable by finitely supported measures

Construction of spaces with arbitrarily high sequential complexity

Sequential closure is always countably obtained in the studied spaces

Abstract

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.