Editing Henri Lebesgue (section)

==Mathematical career==
[[File:Lebesgue - Leçons sur l'integration et la recherche des fonctions primitives, 1904 - 3900788.tif|thumb|''Leçons sur l'integration et la recherche des fonctions primitives'', 1904]]
Lebesgue's first paper was published in 1898 and was titled "Sur l'approximation des fonctions". It dealt with [[Weierstrass]]'s 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 integral]]s 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.<ref name="frs"/>

His lectures from 1902 to 1903 were collected into a "[[Émile Borel|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 [[Augustin-Louis Cauchy]], [[Peter Gustav Lejeune Dirichlet]], and [[Bernhard Riemann]]. Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is "If the sequence f<sub>n</sub>(x) increases to the limit f(x), the integral of f<sub>n</sub>(x) tends to the integral of f(x)." Lebesgue shows that his conditions lead to the [[measure theory|theory of measure]] and [[measurable function]]s and the analytical and geometrical definitions of the integral.

He turned next to [[trigonometry|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 n<sup>th</sup> 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 constant]]s.

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 [[Émile Borel]] about whose integral was more general.<ref>{{cite book|last=Pesin|first=Ivan N.| title=Classical and Modern Integration Theories| year=2014| page=94|publisher=[[Academic Press]]|editor1-last=Birnbaum|editor1-first=Z. W.|editor2-last=Lukacs|editor2-first=E.|url=https://books.google.com/books?id=vr7iBQAAQBAJ&pg=PA94|quote=Borel's assertion that his integral was more general compared to Lebesgue's integral was the cause of the dispute between Borel and Lebesgue in the pages of ''Annales de l'École Supérieure'' '''35''' (1918), '''36''' (1919), '''37''' (1920)|isbn=9781483268699}}</ref><ref>{{cite journal|last=Lebesgue|first=Henri|title=Remarques sur les théories de la mesure et de l'intégration|journal= Annales Scientifiques de l'École Normale Supérieure|year=1918|volume=35|pages=191–250|doi=10.24033/asens.707|url=http://archive.numdam.org/article/ASENS_1918_3_35__191_0.pdf |archive-url=https://web.archive.org/web/20090916213656/http://archive.numdam.org/article/ASENS_1918_3_35__191_0.pdf |archive-date=2009-09-16 |url-status=live|doi-access=free}}</ref><ref>{{cite journal|last=Borel|first=Émile|title=L'intégration des fonctions non bornées|journal= Annales Scientifiques de l'École Normale Supérieure|year=1919|volume=36|pages=71–92|doi=10.24033/asens.713|url=http://archive.numdam.org/article/ASENS_1919_3_36__71_0.pdf |archive-url=https://web.archive.org/web/20140805233602/http://archive.numdam.org/article/ASENS_1919_3_36__71_0.pdf |archive-date=2014-08-05 |url-status=live|doi-access=free}}</ref><ref>{{cite journal|last=Lebesgue|first=Henri|title=Sur une définition due à M. Borel (lettre à M. le Directeur des Annales Scientifiques de l'École Normale Supérieure)|journal= Annales Scientifiques de l'École Normale Supérieure|year=1920|volume=37|pages=255–257|doi=10.24033/asens.725|url=http://archive.numdam.org/article/ASENS_1920_3_37__255_0.pdf |archive-url=https://web.archive.org/web/20090916213617/http://archive.numdam.org/article/ASENS_1920_3_37__255_0.pdf |archive-date=2009-09-16 |url-status=live|doi-access=free}}</ref> 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. These have important practical implications for fundamental physics of which Lebesgue would have been completely unaware, as noted below.