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Henri Léon Lebesgue (French pronunciation: [ɑ̃ʁi leɔ̃ ləˈbɛɡ]; June 28, 1875 – July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation Intégrale, longueur, aire ("Integral, length, area") at the University of Nancy during 1902.

Personal life

Lebesgue's father was a typesetter, who died of tuberculosis when his son was still very young, and Lebesgue himself suffered from poor health throughout his life. After the death of his father, his mother worked tirelessly to support him. He was a brilliant student in primary school, and he later studied at the École Normale Supérieure.

Lebesgue married the sister of one of his fellow students, and he and his wife had two children, Suzanne and Jacques. He worked on his dissertation while teaching in Nancy at a preparatory school.

Mathematical career

Lebesgue's first paper was published in 1898 and was titled "Sur l'approximation des fonctions". It dealt with Weierstrass' theorem on approximation to continuous functions by polynomials. Between March 1899 and April 1901 Lebesgue published six notes in Comptes Rendus. The first of these, unrelated to his development of Lebesgue integration, dealt with the extension of Baire's theorem to functions of two variables. The next five dealt with surfaces applicable to a plane, the area of skew polygons, surface integrals of minimum area with a given bound, and the final note gave the definition of Lebesgue integration for some function f(x). Lebesgue's great thesis, Intégrale, longueur, aire, with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops the theory of measure (see Borel measure). In the second chapter he defines the integral both geometrically and analytically. The next chapters expand the Comptes Rendus notes dealing with length, area and applicable surfaces. The final chapter deals mainly with Plateau's problem. This dissertation is considered to be one of the finest ever written by a mathematician.[1]

His lectures from 1902 to 1903 were collected into a "Borel tract" Leçons sur l'intégration et la recherche des fonctions primitives The problem of integration regarded as the search for a primitive function is the key-note of the book. Lebesgue presents the problem of integration in its historical context, addressing Cauchy, Dirichlet, and Riemann. Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is "If the sequence fn(x) increases to the limit f(x), the integral of fn(x) tends to the integral of f(x)." Lebesgue shows that his conditions lead to the theory of measure and measurable functions and the analytical and geometrical definitions of the integral.

He turned next to trigonometric functions with his 1903 paper "Sur les séries trigonométriques". He presented three major theorems in this work: that a trigonometrical series representing a bounded function is a Fourier series, that the nth Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series is integrable term by term. In 1904-1905 Lebesgue lectured once again at the Collège de France, this time on trigonometrical series and he went on to publish his lectures in another of the "Borel tracts". In this tract he once again treats the subject in its historical context. He expounds on Fourier series, Cantor-Riemann theory, the Poisson integral and the Dirichlet problem.

In a 1910 paper, "Représentation trigonométrique approchée des fonctions satisfaisant a une condition de Lipschitz" deals with the Fourier series of functions satisfying a Lipschitz condition, with an evaluation of the order of magnitude of the remainder term. He also proves that the Riemann–Lebesgue lemma is a best possible result for continuous functions, and gives some treatment to Lebesgue constants.

Lebesgue once wrote, "Réduites à des théories générales, les mathématiques seraient une belle forme sans contenu." ("Reduced to general theories, mathematics would be a beautiful form without content.")

In measure-theoretic analysis and related branches of mathematics, the Lebesgue–Stieltjes integral generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

During the course of his career, Lebesgue also made forays into the realms of complex analysis and topology. He also had a disagreement with Borel (called the teilweise heftig) with regards to effective calculation. However, these minor forays pale in comparison to his contributions to real analysis; his contributions to this field had a tremendous impact on the shape of the field today and his methods have become an essential part of modern analysis.

Lebesgue's theory of integration

This is a non-technical treatment from a historical point of view; see the article Lebesgue integration for a technical treatment from a mathematical point of view.

Integration is a mathematical operation that corresponds to the informal idea of finding the area under the graph of a function. The first theory of integration was developed by Archimedes in the third century BC with his method of quadratures, but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz independently discovered the idea that integration was roughly the inverse operation of differentiation, a way of measuring how quickly a function changed at any given point on the graph. This allowed mathematicians to calculate a broad class of integrals for the first time. However, unlike Archimedes' method, which was based on Euclidean geometry, Newton's and Leibniz's integral calculus did not have a rigorous foundation.

In the nineteenth century, Augustin Cauchy finally developed a rigorous theory of limits, and Bernhard Riemann followed up on this by formalizing what is now called the Riemann integral. To define this integral, one fills the area under the graph with smaller and smaller rectangles and takes the limit of the sums of the areas of the rectangles at each stage. For some functions, however, the total area of these rectangles does not approach a single number. As such, they have no Riemann integral.

Lebesgue invented a new method of integration to solve this problem. Instead of using the areas of rectangles, which put the focus on the domain of the function, Lebesgue looked at the codomain of the function for his fundamental unit of area. Lebesgue's idea was to first build the integral for what he called simple functions, measurable functions that take only finitely many values. Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.

Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree. But there are many functions with a Lebesgue integral that have no Riemann integral.

As part of the development of Lebesgue integration, Lebesgue invented the concept of measure, which extends the idea of length from intervals to a very large class of sets, called measurable sets (so, more precisely, simple functions are functions that take a finite number of values, and each value is taken on a measurable set). Lebesgue's technique for turning a measure into an integral generalises easily to many other situations, leading to the modern field of measure theory.

The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the improper Riemann integral to measure functions whose domain of definition is not a closed interval. The Lebesgue integral integrates many of these functions (always reproducing the same answer when it did), but not all of them. For functions on the real line, the Henstock integral is an even more general notion of integral (based on Riemann's theory rather than Lebesgue's) that subsumes both Lebesgue integration and improper Riemann integration. However, the Henstock integral depends on specific ordering features of the real line and so does not generalise to allow integration in more general spaces (say, manifolds), while the Lebesgue integral extends to such spaces quite naturally.

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Henri Lebesgue