Chapter 24 - Generalized Riemann Integrals

Abstract

This chapter considers the generalized Riemann integrals for Banach space valued functions along with the study of Henstock integral, a slight generalization of the Riemann integral. It is similar to the Riemann integral, but in its power it is more like the Lebesgue integral. The integral studied in this chapter is also known by other names, for example, the special Denjoy integral or the Denjoy-Perron integral, because it is equivalent to a more complicated integral worked out earlier by Denjoy and Perron. The Henstock integral is sometimes known as the generalized Riemann integral. It is also known as the Kurzweil integral or the Henstock-Kurzweil integral, although that last term also has another meaning. It was introduced independently at about the same time by Kurzweil and Henstock. Kurzweil used it as a tool in the study of certain kinds of differential equations particularly for Kurzweil. Henstock developed it in greater detail as part of a wider study of integration theory.