Boolean measure spaces

Abstract

A Boolean measure space is a Boolean σ-space X together with a normalized measure on the σ-field of Borel sets in X, such that non-empty open sets have positive measure and nowhere dense Borel sets have measure zero. The last condition is a very strange one. At first glance it might seem that since a nowhere dense set is topologically small and a set of measure zero is measure-theoretically small, it is fitting and proper that the one should imply the other. A little measure-theoretic experience (with Lebesgue measure in Euclidean spaces, for instance) shows, however, that the implication is not at all likely to hold. The results of this section will show that Boolean measure spaces, in which the implication is assumed to hold, have rather pathological and almost paradoxical properties. The reason for considering them anyway is that measure algebras are important, and, as it turns out, Boolean measure spaces are exactly the duals of measure algebras.