Measures on Boolean Algebras

Abstract

This thesis deals with a number of related results on Boolean algebras. First, we prove the Stone Representation Theorem, which shows that every Boolean algebra is isomorphic to an algebra of sets, namely the clopen algebra of its Stone space. Then we prove the Loomis-Sikorski Theorem, which shows exactly how the Stone Representation Theorem may be extended to represent countable suprema and infima in terms of unions and intersections of sets. Finally, we discuss strictly positive measures. We provide a characterisation, in terms of intersection numbers and covering numbers, of those Boolean algebras which admit strictly positive measures, and we conclude by showing that a σ-complete Boolean algebra admits a strictly positive σ-additive measure if and only if it admits a strictly positive measure and it is weakly σ-distributive.