$\mathfrak G$: 0–1 Finitely Additive Measures

Abstract

This chapter is devoted to the set $\mathfrak G$ of finitely additive measures $\omega$ which take only the values 0 or 1 and explains the sense in which every essentially bounded function is constant $\omega$-almost everywhere. The existence of elements of $\mathfrak G$ with prescribed properties is established using Zorn’s lemma and a relation between elements of $\mathfrak G$ and maximal filters. This observation and its consequences dominate subsequent developments.