Interval Functions and their Integrals, by Ralph Henstock, Ph.D. thesis, 1948

TL;DR

Henstock's 1948 thesis introduces a unified axiomatic approach to non-absolute integration, emphasizing the role of Riemann sum selection, which influenced later developments in gauge integrals and non-Lebesgue integration theories.

Contribution

The thesis presents a novel set of axioms for constructing various integration systems, highlighting common features and unifying different approaches through Riemann sum selection methods.

Findings

Introduced axioms for systems of integration

Unified various non-absolute integration methods

Emphasized Riemann sum selection as key feature

Abstract

Ralph Henstock (1923 - 2007) worked in non-absolute integration, including the Riemann-complete or gauge integral which, independently, Jaroslav Kurzweil also discovered in the 1950's. As a Cambridge undergraduate Henstock took a course of lectures, by J.C. Burkill, on the integration of interval functions. Later, under the supervision of Paul Dienes in Birkbeck College, London, he undertook research into the ideas of Burkill (interval function integrands) and Dienes (Stieltjes integrands); and he presented this thesis in December 1948. The thesis contains the germ of Henstock's later work, in terms of overall approach and methods of proof. For example, a notable innovation is a set of axioms for constructing any particular system of integration. This highlights the features held in common by various systems, so that a particular property or theorem can, by a single, common proof, be shown to hold for various kinds of integration. Within this approach, Henstock's thesis places particular emphasis on various alternative ways of selecting Riemann sums, as the primary distinguishing feature of different systems of integration. This idea was central to his subsequent work and achievement. Of interest also are those ideas in the thesis which were effectively abandoned in his subsequent work. In addition to Henstock's own insights at that stage of his work, this thesis provides a good overview of the literature and state of knowledge of integration of the non-Lebesgue kind at that time. These are good enough reasons for transcribing the thesis. Another pressing reason is that the ink and paper of the near 70 years old copy of the thesis in the Archive in the University of Ulster Library in Coleraine (Henstock's personal, annotated copy) are showing signs of deterioration. Pat Muldowney February 2017