Finitely additive measures on Boolean algebras

Abstract

In this article, we conduct a detailed study of \emph{finitely additive measures} (fams) in the context of Boolean algebras, focusing on three specific topics: freeness and approximation, existence and extension criteria, and integration theory. In the first topic, we present a classification of \emph{free} finitely additive measures, that is, those for which the measure of finite sets is zero, in terms of approximation to uniform probability measures. This inspires a weaker version of this notion, which we call the \emph{uniform approximation property}, characterized in terms of freeness and another well-determined type of fams we call \emph{uniformly supported}. In the second topic, we study new criteria for the existence and extension of finitely additive measures. In particular, we provide an extension of the so-called \emph{compatibility theorem} -- which characterizes when two finitely additive measures can be extended -- yielding a precise and compact characterization of when three finitely additive measures can be simultaneously extended, under the assumption that one of them is an ultrafilter. Finally, we study a Riemann-type integration theory on fields of sets with respect to finitely additive measures, allowing us to extend and generalize some classical concepts and results from real analysis, such as Riemann integration over rectangles in $\mathbb{R}^{n}$ and the Jordan measure. We also generalize the extension criteria for fams allowing desired values of integrals of a given set of functions. At the end, we explore the connection between integration in fields of sets and the Lebesgue integration in the Stone space of the corresponding field, where we establish a characterization of integrability in the sense of the Lebesgue-Vitali theorem, which follows as a consequence of our results.