Comments on Charges on the Boolean Algebra of Regular Open Sets

Abstract

This paper has three main results. The first result is an extension theorem for charges on the Boolean algebra of regular open sets of a topological space. The second result is an abstract form of the first result, which we call an abstract extension theorem for charges on Boolean algebras. The third result is a characterization of regular open sets in $R$. The first two results are useful for integration of functions on topological spaces with respect to charges on Boolean algebras of regular open sets. The third result is useful in proving that the Lebesgue measure is not a charge when restricted to the Boolean algebra of regular open sets (see [3])