[Paper Review] A Sequential Approach to the Henstock Integral

Laramie Paxton|arXiv (Cornell University)|Sep 18, 2016

Mathematical and Theoretical Analysis13 references3 citations

TL;DR

This paper introduces and rigorously develops the Sequential Henstock integral—a novel, sequence-based definition of the Henstock integral over compact intervals in ℝ. It establishes equivalence with the standard ε-δ, Darboux, and topological definitions, proves fundamental theorems including two convergence theorems, and positions the sequential approach as a foundation for extending Henstock integration to abstract spaces and improving calculus pedagogy.

ABSTRACT

The theory of integration over R is rich with techniques as well as necessary and sufficient conditions under which integration can be performed. Of the many different types of integrals that have been developed since the days of Newton and Leibniz, one relative newcomer is that of the Henstock integral, aka the Henstock-Kurzweil integral, Generalized Riemann integral, or gauge integral, which was discovered independently by Henstock and Kurzweil in the mid-1950s. In this paper, we develop an alternative, sequential definition of the Henstock integral over closed intervals in R that we denote as the Sequential Henstock integral. We show its equivalence to the standard epsilon-delta definition of the Henstock integral as well as to the Darboux definition and to a topological definition of the Henstock integral. We then establish the basic properties and fundamental theorems, including two convergence theorems, for the Sequential Henstock integral and offer suggestions for further study.

Motivation & Objective

• To develop a rigorous, sequence-based definition of the Henstock integral, termed the Sequential Henstock integral.
• To demonstrate equivalence between the Sequential Henstock integral and the standard ε-δ, Darboux, and topological definitions of the Henstock integral.
• To establish the basic properties and fundamental theorems—particularly two convergence theorems—for the Sequential Henstock integral.
• To explore the potential of the sequential approach for extending Henstock integration beyond ℝ into abstract topological and metric spaces.
• To assess the pedagogical viability of the Sequential Henstock integral in introductory calculus courses.

Proposed method

• Define the Sequential Henstock integral using generalized sequences (nets) as the core mechanism, replacing the traditional gauge function approach.
• Prove equivalence between the sequential definition and the standard ε-δ definition of the Henstock integral using properties of compact intervals and Cauchy sequences.
• Establish the equivalence with the Darboux definition by showing that the sequential integral satisfies the Darboux criterion for integrability.
• Use topological arguments and completeness of ℝ to show that the sequential definition aligns with the topological characterization of the Henstock integral.
• Apply the theory to prove two convergence theorems for the Sequential Henstock integral, analogous to the dominated and bounded convergence theorems.
• Leverage the sequential framework to suggest generalizations to infinite intervals, ℝⁿ, metric spaces, and topological vector spaces via generalized nets.

Experimental results

Research questions

• RQ1 Is the Sequential Henstock integral equivalent to the standard ε-δ definition of the Henstock integral on compact intervals in ℝ?
• RQ2 Can the Sequential Henstock integral be shown to be equivalent to the Darboux and topological definitions of the Henstock integral?
• RQ3 What fundamental theorems, particularly convergence theorems, can be established for the Sequential Henstock integral?
• RQ4 Can the sequential framework be used to extend the Henstock integral to abstract spaces such as metric spaces or topological vector spaces?
• RQ5 Does the Sequential Henstock integral offer pedagogical advantages for teaching integration in introductory calculus courses?

Key findings

• The Sequential Henstock integral is mathematically equivalent to the standard ε-δ definition of the Henstock integral on compact intervals in ℝ.
• The Sequential Henstock integral is also equivalent to the Darboux and topological definitions of the Henstock integral, establishing its robustness across formulations.
• Two convergence theorems are proven for the Sequential Henstock integral, supporting its use in limit-passing operations.
• The sequential framework provides a natural pathway for generalizing the Henstock integral to infinite intervals and higher-dimensional spaces like ℝⁿ.
• The approach offers a promising foundation for extending Henstock integration to abstract topological and metric spaces using generalized sequences (nets).
• The theory supports the hypothesis that the Sequential Henstock integral may improve student understanding of integration in introductory calculus courses.