Measure and Integration on Boolean Algebras of Regular Open Subsets in a Topological Space

Abstract

The regular open subsets of a topological space form a Boolean algebra, where the join of two regular open sets is the interior of the closure of their union. A content is a finitely additive measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such contents. We then explain the relationship between contents, residual charges, and Borel measures. We show that a content can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the measure of their common boundary. In particular, a content on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how contents can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets.