The Calculus and Gauge Integrals, by Ralph Henstock

TL;DR

This paper discusses the development of the Riemann-complete integral by Ralph Henstock, introducing a novel approach to limits under the integral sign and extending classical convergence theorems within this framework.

Contribution

It presents a new perspective on limits under the integral sign and generalizes classical convergence theorems using the Henstock integral approach.

Findings

Introduces a novel method for limits under the integral sign.

Extends Lebesgue's convergence theorems to the Henstock integral context.

Provides foundational material for a book on calculus and gauge integrals.

Abstract

Ralph Henstock (1923 - 2007), with (independently) Jaroslav Kurzweil, was the originator of the Riemann-complete or generalized Riemann integral. This material consists of four chapters of a book proposal to Cambridge University Press, which remained unpublished in the Henstock Archive at the University of Ulster. Section 5 presents a novel approach to taking limits under the integral sign, and goes on to place the traditional monotone and dominated convergence theorems of Lebesgue in this new context.