Henri Lebesgue

Henri Léon Lebesgue (28 June 1875 – 26 July 1941) was a French mathematician best known for creating a more general theory of integration, expanding on the 17th-century idea of measuring the area between a curve and its axis. He first presented this work in his 1902 doctoral dissertation, *Intégrale, longueur, aire* ("Integral, length, area"), completed at the University of Nancy.

Lebesgue was born in Beauvais, Oise. His father worked as a typesetter, and his mother was a schoolteacher. They built a home library that young Henri used. His father died of tuberculosis when Henri was quite young, leaving his mother to support him alone. Recognizing his strong mathematical talent in primary school, a teacher arranged community funding so he could continue his studies at the Collège de Beauvais, then at Lycée Saint-Louis and Lycée Louis-le-Grand in Paris. In 1894, he entered the École Normale Supérieure, where he concentrated on mathematics, graduating in 1897. He stayed at the school for two more years working in the library, where he encountered the research on discontinuity being done by René-Louis Baire, a recent graduate. Simultaneously, he began graduate work at the Sorbonne, learning about Émile Borel's early measure theory and Camille Jordan's Jordan measure. In 1899, he took a teaching post at the Lycée Central in Nancy while continuing his doctorate. He earned his PhD from the Sorbonne in 1902 with his seminal thesis, advised by Borel. Lebesgue married the sister of a fellow student; they had two children, Suzanne and Jacques.

After his thesis, Lebesgue took a position at the University of Rennes in 1902, lecturing there until 1906, when he moved to the Faculty of Sciences at the University of Poitiers. In 1910, he became a *maître de conférences* at the Sorbonne, becoming a full professor in 1919. He left the Sorbonne in 1921 to become a mathematics professor at the Collège de France, where he taught and researched for the rest of his life. He was elected to the Académie des Sciences in 1922. He died in Paris on 26 July 1941.

Lebesgue's first paper, published in 1898, dealt with Weierstrass's theorem on polynomial approximation of continuous functions. Between March 1899 and April 1901, he published six notes in *Comptes Rendus*. The first extended Baire's theorem to functions of two variables; the next five covered surfaces applicable to a plane, the area of skew polygons, surface integrals of minimum area with a given bound, and the final note defined his integration for a function f(x). His major thesis, *Intégrale, longueur, aire*, appeared in the *Annali di Matematica* in 1902. Its first chapter develops measure theory; the second defines the integral both geometrically and analytically. Later chapters expand his notes on length, area, and applicable surfaces, and the final chapter mainly addresses Plateau's problem. The dissertation is regarded as one of the finest ever written by a mathematician.

His 1902–1903 lectures were collected into a "Borel tract," *Leçons sur l'intégration et la recherche des fonctions primitives*. The book treats integration as the search for a primitive function, presenting the problem historically through Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann. Lebesgue lists six desirable conditions for an integral, the last being that if a sequence fn(x) increases to a limit f(x), then the integral of fn(x) tends to the integral of f(x). He shows these conditions lead to measure theory, measurable functions, and the analytical and geometrical definitions of the integral.

He then turned to trigonometric functions in a 1903 paper, "Sur les séries trigonométriques," presenting three major theorems: a bounded trigonometrical series is a Fourier series, the nth Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and a Fourier series can be integrated term by term. In 1904–1905, he lectured at the Collège de France on trigonometrical series, publishing those lectures in another "Borel tract." There he again treats the subject historically, covering Fourier series, Cantor-Riemann theory, the Poisson integral, and the Dirichlet problem. A 1910 paper, "Représentation trigonométrique approchée des fonctions satisfaisant a une condition de Lipschitz," deals with Fourier series of functions satisfying a Lipschitz condition and evaluates the remainder term's magnitude. He also proves the Riemann–Lebesgue lemma is optimal for continuous functions and discusses 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, the Lebesgue–Stieltjes integral generalizes both Riemann–Stieltjes and Lebesgue integration.