How to Make Measure Theory Completely Pointless

This manuscript explores measure theory from a novel perspective, based on Boolean algebras, and establishes a framework that is independent of any underlying set of points. We provide a preliminary investigation of the benefits of such a framework, by introducing a novel generalization of the concept of a function, and deducing the consequences of this notion for measure theory.

The manuscript is divided into two chapters. The first chapter provides a comprehensive overview of existing results in measure theory on Boolean algebras, drawing from the literature. We then introduce a generalized concept of functions that abstracts the underlying space, providing an important building block for measure theory on Boolean algebras. This construction has significant implications for various topics, including measures, spaces of measures, and conditional distributions.

In the second chapter, we leverage the insights gained from the first chapter to explore the application of these ideas to convex analysis in infinite-dimensional spaces. Inspired by the non-atomic nature of Boolean algebras, we investigate the concept of faces as a generalization of extreme points, which may not exist in non-compact convex sets. We also introduce the notion of full faces, which we believe holds importance in the study of convex analysis, particularly in the context of infinite-dimensional spaces.