A ug 1 99 4 Properties of the Class of Measure Separable Compact Spaces

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as M S. We discuss some closure properties of M S, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in M S. Most of the basic theory for regular measures is true just in ZF C. 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 ≤ c. We show that not being in M S is preserved by all forcing extensions which do not collapse ω 1 , while being in M S can be destroyed even by a ccc forcing. §0. Introduction. As we learn in a beginning measure theory course, every Borel measure on a compact metric space is separable. It is natural to ask to what extent this generalizes to other compact spaces. It is also true that every Borel measure on a compact metric space is regular. In this paper, we study the class, M S, of compacta, X, with the property that every regular measure on X is separable. This contains some simple spaces (such as compact ordered spaces and compact scattered spaces), and has some interesting closure properties. One might also study the class of compacta X such that every Borel measure on X is separable, but the theory here is very sensitive to the axioms of set theory; for example, the existence of an ordered scattered compactum with a non-separable Borel