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

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.

• The paper demonstrates that the Lebesgue integral enables a stronger Dominated Convergence Theorem, crucial for establishing the integrability of limit functions.
• The paper emphasizes the completeness of Lebesgue function spaces, which underpins robust applications in Fourier transform theory and functional analysis.
• The paper outlines how topological nuances and linear limit structures provide practical advantages in mathematical proofs over traditional Riemann integration.

Understanding the Paper's Argument

The paper "Should we fly in the Lebesgue-designed airplane? -- The correct defence of the Lebesgue integral" (2309.08908) addresses the often misunderstood importance of the Lebesgue integral over the Riemann integral. Through mathematical rigor, it clarifies why the Lebesgue integral holds prominence in mathematical theories and applications. Specifically, it highlights the critical distinctions regarding convergence theorems and the completeness of functional spaces that rely on the Lebesgue integral.

Key Differences Between Riemann and Lebesgue Integrals

The paper systematically debunks the common misconception that the Lebesgue integral is merely a generalization of the Riemann integral without significant utility. It argues that key differences lie in the Dominated Convergence Theorems and completeness properties of normed vector spaces associated with these integrals:

1. Dominated Convergence Theorems: The Lebesgue integral supports a more powerful version of the Dominated Convergence Theorem than the Riemann integral. This theorem allows for conclusions about the integrability of limit functions, crucial for many proofs in applied mathematics, such as those involving the completeness of $L^p$-spaces.
2. Completeness of Spaces: The Lebesgue spaces, such as $\hat{L}^1([0,1];R)$, are complete, which is a fundamental property for functional analysis and applications like Fourier transform theory. In contrast, the space $\hat{R}^1([0,1];R)$ is shown to be incomplete, demonstrating a significant deficiency when attempting to utilize Riemann-integrable functions in practical applications.

Implications on Fourier Transform Theory

A particularly illustrative consequence explored in the paper is the application to Fourier transform theory. The inability of Riemann integrable functions to support a well-defined $L^2$-Fourier transform highlights the theoretical and practical limitations of the Riemann integral. Complete spaces like $\hat{L}^2(R;C)$ are essential to sustaining integral transforms which are cornerstones of signal processing and functional analysis.

Topological Interpretations

The paper discusses the topological and limit structures that differentiate the two integration theories. These include:

• Pointwise Convergence Topologies: The paper describes how the Lebesgue integral allows for almost everywhere pointwise convergence, a concept not properly supported by topologies permissible under the Riemann framework.
• Linear Limit Structures: It introduces the notion of limit structures which are necessary to describe convergence properties that are not characterizable by traditional topological concepts alone.

Theoretical and Practical Significance

The insights provided have far-reaching implications both theoretically and practically:

• Mathematically, understanding these properties allows researchers to correctly utilize integration strategies in proofs and applications.
• Practically, when designing algorithms or systems for signal processing, choosing the proper integration theory can lead to more robust and reliable outputs.

Conclusion

This paper provides an authoritative and nuanced perspective on why the Lebesgue integral is vital not merely for its broader applicability but for its stronger mathematical properties that support essential theorems and applications. It urges mathematicians and application-oriented researchers to recognize the integral's importance beyond its generalized scope, especially in fields where integration plays a crucial role in analysis and synthesis.