Properties of the Class of Measure Separable Compact Spaces

TL;DR

This paper studies the class of compact spaces where all regular Borel measures are separable, exploring their properties, closure characteristics, and the set-theoretic conditions affecting their measure-theoretic features.

Contribution

It characterizes the class MS of compact spaces with separable measures, examines its closure properties, and links the existence of certain non-separable measures to large cardinal hypotheses.

Findings

Compact ordered and scattered spaces are in MS.

Non-separable measures relate to real-valued measurable cardinals.

Being in MS can be affected by forcing extensions.

Abstract

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure is equivalent to the existence of a real-valued measurable cardinal less or equal to c. We show that not being in MS is preserved by all forcing extensions which do not collapse omega_1, while being in MS can be destroyed even by a ccc forcing.