[Paper Review] Ralph Henstock's Lectures on the Theory of Integration

Ralph Henstock, Pat Muldowney|arXiv (Cornell University)|Dec 15, 2015

Mathematical and Theoretical Analysis3 references3 citations

TL;DR

This paper presents Ralph Henstock's 1970–71 lectures on integration theory, focusing on the Riemann-complete (gauge) integral in Euclidean space and extending it to an abstract theory of integration. It establishes the equivalence of Riemann and Darboux integrals for real-valued functions and explores the limitations of classical integration when dealing with derivatives of functions that are not absolutely integrable, ultimately laying foundational groundwork for the Henstock–Kurzweil integral and its application to generalized integrals such as the Wiener and Feynman integrals.

ABSTRACT

These are the class notes of lectures given by Ralph Henstock at the New University of Ulster in 1970-71. The notes deal with the Riemann-complete integral (also known as the generalized Riemann integral, the gauge integral, and the Henstock-Kurzweil integral). They also introduce Henstock's abstract theory of integration.

Motivation & Objective

• To clarify and systematize the theory of the Riemann-complete (gauge) integral in $\mathbb{R}^n$, offering a clearer exposition than earlier works.
• To establish the equivalence between Riemann and Darboux integration for real-valued functions, reinforcing foundational consistency.
• To extend the integration framework beyond the Riemann integral to handle functions with unbounded variation or non-absolutely integrable derivatives, such as those arising in calculus and differential equations.
• To develop an abstract theory of integration (Henstock integral) applicable to generalized integrals, including Wiener and Feynman integrals, despite challenges in Fubini-type theorems.
• To investigate the applicability of limit theorems in integration, particularly whether the integral of a limit equals the limit of integrals, in the context of generalized integrators.

Proposed method

• Uses the Riemann sum $ R(f;\mathcal{D}) = \sum f(P)\mu(J) $ over divisions $ \mathcal{D} $ of a brick $ I $, with norm $ \|\mathcal{D}\| \to 0 $, to define the Riemann integral as the limit of these sums.
• Applies Darboux sums $ S(f;\mathcal{D}) = \sum M(f;J)\mu(J) $ and $ s(f;\mathcal{D}) = \sum m(f;J)\mu(J) $, where $ M(f;J) $ and $ m(f;J) $ are the supremum and infimum of $ f $ on $ J $, to define the Darboux integral.
• Demonstrates equivalence between Riemann and Darboux integrals via the squeeze theorem: if $ s(f;\mathcal{D}) $ and $ S(f;\mathcal{D}) $ converge to the same limit $ r $, then so does $ R(f;\mathcal{D}) $.
• Introduces the concept of variationally equivalent integrators, showing that expressions like $ p(x_1,\ldots,x_n;C)\Delta x_1\cdots\Delta x_n $ can replace $ P(I) $ in integration if the integrator is continuous and positive.
• Applies continuity and positivity of the Wiener density function $ w(x_1,\ldots,x_n;C) $ to construct neighborhoods where the integrand remains close to its value, enabling approximation via Riemann sums.
• Analyzes the Feynman integrator by considering $ |p| = \pi^{-n/2} s^{-1} $, and shows that under certain conditions, the variation of the integrand over sets of small measure tends to zero, implying almost everywhere convergence of sequences of functions.

Experimental results

Research questions

• RQ1 Under what conditions is the Riemann integral equivalent to the Darboux integral for real-valued functions?
• RQ2 Can the Riemann-complete integral integrate functions that are not Lebesgue integrable, such as derivatives of functions with unbounded variation?
• RQ3 To what extent can the abstract Henstock integral framework be applied to generalized integrals like the Wiener and Feynman integrals?
• RQ4 Does the limit theorem—integral of the limit equals the limit of the integrals—hold for the Henstock integral in the context of Feynman integrands?
• RQ5 Why does Fubini’s theorem fail in the case of the Wiener integrator, and can it still be applied meaningfully in practice?

Key findings

• The Riemann and Darboux integrals are equivalent for real-valued functions, as shown by the squeeze theorem applied to upper and lower sums.
• The Riemann integral exists only for bounded functions, as unboundedness leads to undefined Darboux sums.
• The calculus integral (antiderivative-based) can integrate functions whose derivatives are not Riemann or Lebesgue integrable, such as $ F'(x) = x^{-1/2} $ on $[0,1]$, where $ F(x) = 2x^{1/2} $.
• The Wiener integrator $ w(x_1,\ldots,x_n;C) $ is continuous and positive, allowing the construction of neighborhoods where the integrand remains uniformly close to its value, enabling variational equivalence of integrators.
• The Feynman integrator does not satisfy the VBG* condition, but despite this, the theory remains applicable to 'very nice' functions like $ \sin $ and $ \cos $, and limit theorems can still be applied in practice.
• For sequences $ F_m \to F $, if the variation of $ (F_m - F)p\Delta x_1\cdots\Delta x_n $ over small sets tends to zero, then $ F_m = F $ almost everywhere, supporting convergence in the Henstock sense.