RALPH HENSTOCK'S INFLUENCE ON THE INTEGRATION THEORY

Abstract

Professor Ralph Henstock, a distinguished analyst who devoted most of his research to integration theory, died on January 7, 2007. It was my good fortune to know Ralph for almost 30 years, and to derive much inspiration from his pioneering work. Rather than commenting briefly on a variety of Ralph’s achievements, I will elaborate on that which has been the most influential: a Riemann type definition of the Denjoy-Perron integral . An early recognition that the powerful Lebesgue integral does not integrate the derivatives of all differentiable functions led to the development of the Denjoy-Perron integral. Three equivalent definitions of the Denjoy-Perron integral were available at the beginning of the last century: descriptive and constructive definitions presented by Denjoy [16, Chapter 8], and a definition based on approximations by majorants and minorants due to Perron [16, Chapter 6, Section 6]. These definitions differ widely, and establishing their equivalence is not easy — for a comprehensive treatment see [4, Chapter 11]. Neither definition is simple, and attempts to generalize any of them to higher dimensions were not successful. Indeed, none of the multidimensional integrals based on these definitions integrates partial derivatives of all differentiable functions. It was a major independent accomplishment of Henstock [5] and Kurzweil [10] to observe that a minor but ingenious change in the classical definition of the Riemann integral produces the Denjoy-Perron integral. The striking simplicity of their definition revitalized the efforts toward finding a multidimensional analog of the Denjoy-Perron integral. The initial impetus was further promoted by Henstock’s diligent work on the general properties of Riemann type integrals, summarized in monographs [6, 7, 8]. After introducing the Henstock-Kurzweil result without proof, I will discuss two separate but related topics: