Henri Lebesgue and the Development of the Integral Concept: A Mini-Primary Source Project for Undergraduate Analysis Students

What classes of functions are integrable? The full answer to this question—that it depends on the type of integration used—is an intriguing fact not generally known to our introductory analysis students.

In the earliest days of calculus, the process of integration was considered to be identical with that of finding an antiderivative. While this meant that any expression that was not itself a derivative could not be integrated, this limitation was neither a theoretical or practical concern at that time. Today, we know that antidifferentiable functions necessarily possess the Intermediate Value Property. This result, known as Darboux's Theorem, rules out integrability for basic step functions under the “integration as antidifferentiation” conception. On the other hand, these discontinuous functions are quite easily integrated under the geometric view of “area under the curve” that emerged as the primary conception of the integral over the course of the 18th century.

Lebesgue's diagram for the Riemann integral.

It was this geometric view of an integral that Cauchy sought to capture by defining the integral as the limit of what is today called a Riemann sum, in tribute to Riemann's thorough investigation of conditions for integrability under that definition. But as Lebesgue noted in the introduction to his doctoral dissertation [Lebesgue 1902]:

It is known that there are derivatives that are not integrable, if one accepts Riemann's definition of the integral; the kind of integration as defined by Riemann does not allow in all cases to solve the fundamental problem of calculus:

Find a function with a given derivative.

It thus seems natural to search for a definition of the integral which makes integration the inverse operation of differentiation in as large a range as possible.

The outcome of Lebesgue's search for such a definition was the integral named in his honor.

The mini-Primary Source Project (PSP) Henri Lebesgue and the Development of the Integral Concept uses excerpts from the relatively non-technical paper “Sur le développement de la notion d’intégrale” [Lebesgue 1927] as a means to consolidate students' understanding of the Riemann integral and its relative strengths and weaknesses. Following a brief overview of the evolution of the integral concept, Lebesgue contrasted the conceptual notions behind the Riemann and Lebesgue integrals. Lebesgue's clever metaphorical comparisons of Riemann's approach to that of an unsystematic merchant “who counts coins and bills at random in the order in which they came to hand,” in contrast to his own approach to that of “a methodical merchant” who stacks coins of the same denomination together before counting them, allowed him to quite naturally bring in the concept of set measure and its role in defining his integral.

Diagram from [Lebesgue 1927] showing the set \(E_i= \{ x \, \vert \, y_i\le f(x) \le y_{i+1}\}\).

In keeping with the content objectives of a typical undergraduate introductory analysis course, the PSP's primary content focus is on the definition and properties of the Riemann integral. Reading about Lebesgue's motivations for developing a different type of integration in his own words also allows students to witness the ways in which mathematicians hone various tools of their trade (e.g., definitions, theorems). The project further touches on issues related to the tensions between “logical rigor” and “geometrical intuition” as guiding principles in mathematics. In fact, Lebesgue explicitly described his new definition of the integral as an effort to reconcile these two desirable but conflicting aspects of mathematics. Additionally, this project offers undergraduates a first (if brief) glimpse of the integral that is the current standard in graduate courses and mathematical research—at least for the time being!

The complete project Henri Lebesgue and the Development of the Integral Concept (pdf) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explains the purpose of the project and offers guidance on its implementation is appended at the end of the student project.

This project is the seventh in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes thirteen PSPs for use in a real analysis course.

Acknowledgments

The development of the student project Henri Lebesgue and the Development of the Integral Concept has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation's Improving Undergraduate STEM Education Program under grant number 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily represent the views of the National Science Foundation. The author also wishes to thank George W. Heine III for recreating the diagrams used in this article.

References

Lebesgue, Henri. 1902. Intégrale, Longueur, Aire. PhD thesis, Université de Paris. Milan: Bernandon de C. Rebeschini.

Lebesgue, Henri. 1927. Sur le développement de la notion d’intégrale. Revue de Metaphysique et de Morale, 34(2):149–167. English translation by Kenneth O. May. In Classics of Mathematics, edited by R. Calinger, 762–765. Prentice-Hall, 1995.