Measure and Integration

Abstract

In this chapter and the next, the theory of integration in uniform spaces will be developed. This chapter will only be concerned with those aspects of integration theory that do not depend on the uniform structure of the space. In elementary analysis one encounters the concept of the Riemann integral. Intuitively, the process of Riemann integration in one, two and three dimensional Euclidean space corresponds to calculating lengths, areas and volumes respectively. The formalization of the Riemann integral occurred during the nineteenth century. Briefly, the main idea for one dimensional Euclidean space is that the Riemann integral of a function f over an interval [a,b] can be approximated by sums of the form $$ \sum\nolimits_i^n = 1f(x_i )\Delta (I_i ) $$ where I1 … I n are disjoint intervals whose union is [a, b], Δ(I i ) denotes the length of I i and x i ∈ I i for each i = 1…n.