In the first decade of the 20th century, Henri Lebesgue developed a new approach to integral calculus in order to overcome the restrictions of previous theories. At that time, integration was used to calculate the area under a curve, but if the curve was discontinuous the theory was difficult to apply and left some questions unanswered. Lebesgue's theory of integration circumvented the problems caused by these discontinuities and was compatible with other basic mathematical operations.
Henri Léon Lebesgue was born in Beauvais, France, on June 28, 1875. His father was a typographical worker, and his mother was an elementary school teacher. He entered the École Normale Supérieure in 1894 and quickly demonstrated mathematical talent along with an irreverent attitude that gave him the tendency to ignore subjects that did not interest him. For instance, he passed his chemistry course only by mumbling his answers to the examiner who was hard of hearing. Even in mathematics he graduated third in his class in 1879. Nevertheless, his questioning of the traditional methods of mathematics was the basis for his reexamination of the concepts of length, area, and volume. He stayed on to work in the library for two years after his graduation in 1879.
Lebesgue inherited the solid foundation for the theory of calculus that was laid by the mathematical giants of the 19th century. Karl Friedrich Gauss, Augustin-Louis Cauchy , Niels Henrik Abel , and others of this period had introduced rigorous definitions of convergence, limit, and continuity. They had also formulated a precise definition of the integral, one of the two central concepts of calculus. Just as addition gives the total of a finite set of numbers, the integral pertains to the limiting case of the sum of a quantity that varies at every one of an infinite set of points. Such a quantity is described mathematically by a function, and integration of a function can represent the area bounded by a curve, the total work done by a variableforce, the distance a planet travels in its elliptical orbit around the Sun, among other possibilities. Cauchy's definition of the integral applied to functions that were continuous, that is, curves without any jumps. It could also handle a finite number of discontinuities, points where jumps occurred.
In 1854 Georg Riemann introduced an extension of the concept of integration which found its way into most calculus books. Unfortunately, the Riemann integral was unsatisfactory for dealing with some sequences of functions. Even if the sequence of functions approached a limit and each function in the sequence was continuous, the limit might not be a function that could be handled by Riemann integration. The problem was to find a definition for integration that would be compatible with taking the limit of a sequence of functions.
Builds Upon Previously Established Theories
Lebesgue was influenced by the work of René Baire (another recent graduate of the École Normale Supérieure) and Émile Borel . By 1898, when Lebesgue published his first results, Baire had formulated an insightful theory of discontinuous functions. In that same year Borel published a theory of measure that generalized the concepts of area to new types of regions obtained as limits.
Lebesgue taught at the Lycée Central in Nancy from 1899 to 1902. During that time he developed the ideas for his doctoral thesis at the Sorbonne. The work, "Intégrale, longueur, aire," extended Borel's theory of measure, defined the integral geometrically and analytically, and established nearly all the basic properties of integration. J. C. Burkill notes in the obituary of Lebesgue for the Journal of the London Mathematical Society: "It cannot be doubted that this dissertation is one of the finest which any mathematician has ever written."
Lebesgue's consideration of discontinuous curves and nonsmooth surfaces was shocking to some of his contemporaries. Camille Jordan cautioned Lebesgue that he should not expect other scholars to appreciate his work. Fortunately, the usefulness of his new ideas quickly overcame any resistance, and Lebesgue received a university appointment as maître des conférences at Rennes in 1902. During his first year he gave lectures for the Cours Peccot at the Collège de France on his new integral and the next year on its application to trigonometric series. Lebesgue published these lectures in the series of tracts edited by Borel and was the first author in this series of monographs other than Borel himself. He gave a thorough exposition of the historical background of the problems leading up to the properties an integral should satisfy, including the compatibility with the limit of a sequence of functions.
In 1906 Lebesgue left Rennes to become chargé de cours for the faculty of sciences, and later a professor, at Poitiers. In 1910 he was maître des conférences at the Sorbonne, and in 1921 he became professor at the Collège de France. Among his many prizes and honors was his election to the Académie des Sciences in 1922. By that time Lebesgue had nearly 90 publications on measure theory, integration, geometry, and related topics. Although his ideas were ignored at the great centers of mathematics such as Göttingen, his integral was presented to undergraduates at Rice Institute as early as 1914, and served as an inspiration to the founders of the Polish schools of mathematics at Lvov and Warsaw in 1919.
During the last 20 years of his life, Lebesgue's work became widely known, and his approach to integration evolved as a standard tool of analysis. Lebesgue himself began to concentrate more on the historical and pedagogical issues associated with his work. He believed that mathematicians should work from the problems that motivate theory and resist being bound to tradition. He felt that mathematical education should follow this same principle. He freely used the words "deception" and "hypocrisy" to describe the lack of connection between students' natural intuition of numbers and geometry and the manner in which these subjects were taught. In "Sur le mesure des grandeurs," Lebesgue complained about teaching of mathematics: "An infinite amount of talent has been expended on little perfections of detail. We must now attempt an overhaul of the whole structure."
Lebesgue died in Paris on July 26, 1941. Even during the last months of his terminal illness, he continued his course on geometrical constructions at the Collège de France and dictated a book on conic sections. He was survived by his wife, mother, a son, and a daughter. His view of mathematics can be summed up in the concluding words of "Sur le développement de la notion d'intégrale": "A generalization made not for the vain pleasure of generalizing but in order to solve previously existing problems is always a fruitful generalization. This is proved abundantly by the variety of applications of the ideas that we have just examined."
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