Integration

Abstract

Using the notions of measure theory introduced in Chap. 1, an integration process is introduced for functions with values in normed vector spaces. Such an extension does not require much supplementary effort but can be bypassed in a first reading. Convergence results and calculus rules form the bulk of the chapter.

In the case of continuous functions, the notion of integral

coincides with the notion of primitive. Riemann has defined the

integral of some discontinuous functions, but not all derivative

functions are integrable in the Riemann sense. Thus,

the problem of searching for primitive functions through integration

is not solved, and one may wish for a definition of an integral

including Riemann’s which allows one to solve the problem of

primitive functions.

Henri Lebesgue, Sur une généralisation de l’intégrale définie,

Comptes-rendus de l’Académie des Sciences de Paris 132,

pp. 128–132, April 29th 1901.