Integrals and measures

Abstract

In the language of modern integration theory the term integral refers to a number of somewhat different concepts, arrived at through a variety of constructions and definitions. About the only thing that can be said about integration in reasonable generality is that an integral on a space X is a linear transformation that is defined on a vector space of functions on X and satisfies certain continuity requirements. As regards the Lebesgue integral, however, matters are in a much less chaotic state. Indeed, while a considerable number of different definitions and constructions can be found in the literature, there is unanimous agreement on what a Lebesgue integral is. We provide an axiomatic characterization.