Should we fly in the Lebesgue-designed airplane? -- The correct defence of the Lebesgue integral

TL;DR

This paper clarifies the fundamental reasons for the Lebesgue integral's importance over the Riemann integral, emphasizing differences in convergence theorems and space completeness, supported by topological insights and counterexamples.

Contribution

It provides a topological interpretation of key differences between Lebesgue and Riemann integrals and illustrates the limitations of the Riemann integral with explicit counterexamples.

Findings

Topological interpretations of the Dominated Convergence Theorem differences

Counterexamples illustrating Riemann integral deficiencies

Implications for Fourier transform theory when restricted to Riemann integrable functions

Abstract

It is well-known that the Lebesgue integral generalises the Riemann integral. However, as is also well-known but less frequently well-explained, this generalisation alone is not the reason why the Lebesgue integral is important and needs to be a part of the arsenal of any mathematician, pure or applied. Those who understand the correct reasons for the importance of the Lebesgue integral realise there are at least two crucial differences between the Riemann and Lebesgue theories. One is the difference between the Dominated Convergence Theorem in the two theories, and another is the completeness of the normed vector spaces of integrable functions. Here topological interpretations are provided for the differences in the Dominated Convergence Theorems, and explicit counterexamples are given which illustrate the deficiencies of the Riemann integral. Also illustrated are the deleterious consequences of the defects in the Riemann integral on Fourier transform theory if one restricts to Riemann integrable functions.