Written by one of the subject's foremost experts, this is the first book on division space integration theory. It is intended to present a unified account of many classes of integrals including the Lebesgue-Bochner, Denjoy-Perron gauge, Denjoy-Hincin, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals. Professor Henstock develops here the general axiomatic theory of Riemann-type integration from first principles in such a way that familiar
classes of integrals (such as Lebesgue and Wiener integrals) are subsumed into the general theory in a systematic fashion. In particular, the theory seeks to place Feynman integration on a secure
analytical footing. By adopting an axiomatic approach, proofs are, in general, simpler and more transparent than have previously appeared. The author also shows how one proof can prove corresponding results for a wide variety of integrals. As a result, this book will be the central reference work in this subject for many years to come.
'Henstock's book develops ideas of great ingenuity and insight in an attempt to fit non-absolute integrals into a theory that, unlike most earlier theories includes many more integrals than the classical one of Denjoy and Perron ... it is a valiant attempt to bring a difficult subject under control and should be known to all interested in integration.'
P.S. Bullen, Zentralblatter 745
'a thorough reference for experts'
Mathematika, 40 (1993)
`A very good and exhaustive list of relevant publications is given in the references. A large amount of topics is treated in this book of R. Henstock. They are explained in a close form which is traditional for the author's style of writing. The book represents a source of inspiration for research in the theory of integration. Surely there will be a wide use of this work as a standard reference in the future for scientists working in the contemporary
summation approach to nonabsolutely convergent integral.'
J. Kurzweil, S. Schwabik, Sonderdruck aus Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd 96, Heft 3
Introduction and prerequisites; Division systems and division spaces; Generalized Riemann and variational integration in division systems and division spaces; Limits under the integral sign, functions depending on a parameter; Differentiation; Cartesian products of a finite number of division systems (spaces); Integration in infinite-dimensional spaces; Perron-type, Ward-type, and convergence-factor integrals; Functional analysis and integration theory;