Overview

Abstract

Divergent processes are the raison d’étre of most of classical and modern analysis. Divergent processes, or rather control and management of such, occur whenever a limit procedure, or combination of, is present. Perhaps the most graphic and at the same time subtle example of this lies in the notion of integration. It is well known that for the Riemann integral of a bounded function on [a, b] to exist, it is necessary as sufficient that the function be continuous almost everywhere. Loosely put, the Riemann integral is basically appropriate for the class of continuous functions on [a, b], C[a, b]. Furthermore, the natural notion of convergence for C[a, b] is that of uniform convergence. The latter fact indicates a convergence theorem: one may interchange a sequential limit operation and the Riemann integral, if the limit operation is uniform. Lebesgue’s idea, introducing the notion of measurable functions, not only led to a broader class of functions which can be integrated while keeping all operational properties of the Riemann integral, but also led to much richer sequential limit theorems. Other examples of substantial importance are as follows.