# Henri Lebesgue

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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.

## Contents

### 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 keynote 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.