Foundations of Functional Analysis

Abstract

We start with the Riemannian integral - and their Riemann integrable functions - and construct a considerably larger class of integrable functions via an extension procedure. Then we obtain Lebesgue’s integral, which is distinguished by general convergence theorems for pointwise convergent sequences of functions. This extension procedure - from the Riemannian integral to Lebesgue’s integral - will be provided by the Daniell integral. The measure theory for Lebesgue measurable sets will appear in this context as the theory of integration for characteristic functions. We shall present classical results from the theory of measure and integration in this chapter, e.g. the theorems of Egorov and Lusin.