It is hardly possible to overemphasize the importance of the theory of
integration to mathematical analysis; indeed, it is one of the twin pillars on
which analysis is built. Granting that, it is surprising that new developments
continue to arise in this theory, which was originated by the great Newton
and Leibniz over three centuries ago, made rigorous by Riemann in the
middle of the nineteenth century, and extended by Lebesgue at the beginning
of the twentieth century.
The purpose of this monograph is to present an exposition of a rela-
tively new theory of the integral (variously called the "generalized Riemann
integral", the "gauge integral", the "Henstock-Kurzweil integral", etc.) that
corrects the defects in the classical Riemann theory and both simplifies and
extends the Lebesgue theory of integration. Not wishing to tell only the
easy part of the story, we give here a complete exposition of a theory of in-
tegration, initiated around 1960 by Jaroslav Kurzweil and Ralph Henstock.
Although much of this theory is at the level of an undergraduate course in
real analysis, we are aware that some of the more subtle aspects go slightly
beyond that level. Hence this monograph is probably most suitable as a text
in a first-year graduate course, although much of it can be readily mastered
by less advanced students, or a teacher may simply skip over certain proofs.
The principal defects in the Riemann integral are several. The most seri-
ous one is that the class of Riemann integrable functions is too small. Even
in calculus courses, one needs to extend the integral by defining "improper
integrals", either because the integrand has a singularity, or because the
interval of integration is infinite. In addition, by taking pointwise limits of
Riemann integrable functions, one quickly encounters functions that are no
Previous Page Next Page