- Henri Lebesgue
Infobox Scientist

name =Henri Lebesgue

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birth_date =1875-06-28

birth_place =Beauvais ,France

death_date =death date and age|1941|7|26|1875|6|28

death_place =Paris ,France

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field =Mathematics

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alma_mater =École Normale Supérieure ,France

doctoral_advisor =Émile Borel

doctoral_students =Paul Montel Zygmunt Janiszewski Georges de Rham

known_for =Lebesgue integration Lebesgue measure

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footnotes =**Henri Léon Lebesgue**IPA| [ɑ̃ʁiː leɔ̃ ləˈbɛg] (June 28 ,1875 ,Beauvais –July 26 ,1941 ,Paris ) was a Frenchmathematician , most famous for his theory of integration. Lebesgue's integration theory was originally published in his dissertation, "Intégrale, longueur, aire" ("Integral, length, area"), at the University ofNancy in 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 theÉcole 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 ofBaire's Theorem to functions of two variables. The next five dealt with surfaces applicable to a plane, the area of skewpolygons ,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 thèse, 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 (seeBorel 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 withPlateau's problem . This dissertation is considered to be one of the finest ever written by a mathematicianJ. C. Burkill, Obituary Notices of Fellows of the Royal Society, Vol. 4, No. 13. (Nov., 1944), pp. 483-490.] .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 , andRiemann . Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is "If the sequence f_{n}(x) increases to the limit f(x), the integral of f_{n}(x) tends to the integral of f(x)". Lebesgue shows that his conditions lead to the theory of measure andmeasurable function s 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 seriesrepresenting a bounded function is a Fourier series, that the n

^{th}Fourier coefficient tends to zero (theRiemann-Lebesgue lemma ), and that aFourier series is integrable term by term. In 1904-1905 Lebesgue lectured once again at theCollège 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, thePoisson integral and theDirichlet 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 theRiemann-Lebesgue lemma is a best possible result for continuous functions, and gives some treatment toLebesgue 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 andtopology . He also had a disagreement with Borel (called theteilweise heftig ) with regards toeffective calculation . However, these minor forays pale in comparison to his contributions toReal analysis ; his contributions to this field had a tremendous impact on the shape of the field today and his methods have become an integral 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 byArchimedes 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 onEuclidean geometry , Newton's and Leibniz'sintegral calculus did not have a rigorous foundation.In the nineteenth century,

Augustin Cauchy finally developed a rigorous theory of limits, andBernhard Riemann followed up on this by formalising what is now called theRiemann integral . To define this integral, one fills the area under the graph with smaller and smallerrectangle s and takes the limit of thesum s 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 calledsimple function s, 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 beautiful 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

Lebesgue measure , which extends the idea oflength from intervals to a very large class of sets, called measurable sets (so, more precisely,simple function s 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 ofmeasure 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 aclosed 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, theHenstock 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 thereal 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.**ee also***

Dominated convergence theorem

*Lebesgue covering dimension

*Lebesgue point

*Lebesgue's number lemma

*Lebesgue spine

*Lebesgue constant (interpolation) **References**1 & 2.

J. C. Burkill , Obituary Notices of Fellows of the Royal Society, Vol. 4, No. 13. (Nov., 1944), pp. 483-490. [*http://links.jstor.org/sici?sici=1479-571X%28194411%294%3A13%3C483%3AHL1%3E2.0.CO%3B2-A*]**External links***

*

* [http://mathweb.free.fr/bios/index.php3?action=affiche&quoi=lebesgueHenri Léon Lebesgue (28 juin 1875 [Rennes] - 26 juillet 1941 [Paris] ) ] "(in French)"**Original articles written by Lebesgue (in French)*** [

*http://archive.numdam.org/article/BSMF_1903__31__197_1.pdf Sur le problème des aires 1*] , 1903

* [*http://archive.numdam.org/article/ASENS_1903_3_20__453_0.pdf Sur les séries trigonométriques*] , 1903

* [*http://archive.numdam.org/article/BSMF_1904__32__229_0.pdf Une propriété caractéristique des fonctions de classe 1*] , 1904

* [*http://archive.numdam.org/article/BSMF_1905__33__273_1.pdf Sur le problème des aires 2*] , 1905

* [*http://archive.numdam.org/article/BSMF_1907__35__202_1.pdf Contribution à l'étude des correspondances de M. Zermelo*] , 1907

* [*http://archive.numdam.org/article/BSMF_1908__36__3_1.pdf Sur la méthode de M. Goursat pour la résolution de l'équation de Fredholm*] , 1908

* [*http://archive.numdam.org/article/AFST_1909_3_1__25_0.pdf Sur les intégrales singulières*] , 1909

* [*http://archive.numdam.org/article/AFST_1909_3_1__119_0.pdf Remarques sur un énoncé dû à Stieltjes et concernant les intégrales singulières*] , 1909

* [*http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1910_3_27_/ASENS_1910_3_27__361_0/ASENS_1910_3_27__361_0.pdf Sur l'intégration des fonctions discontinues*] , 1910

* [*http://archive.numdam.org/article/BSMF_1910__38__184_0.pdf Sur la représentation trigonométrique approchée des fonctions satisfaisant à une condition de Lipschitz*] , 1910

* [*http://archive.numdam.org/article/BSMF_1912__40__238_1.pdf Sur un théorème de M. Volterra*] , 1912

* [*http://archive.numdam.org/article/BSMF_1917__45__132_1.pdf Sur certaines démonstrations d'existence.*] , 1917

* [*http://archive.numdam.org/article/ASENS_1918_3_35__191_0.pdf Remarques sur les théories de la mesure et de l'intégration.*] , 1918

* [*http://archive.numdam.org/article/ASENS_1920_3_37__255_0.pdf Sur une définition due à M. Borel (lettre à M. le Directeur des Annales Scientifiques de l'École Normale Supérieure)*] , 1920

* [*http://archive.numdam.org/article/AFST_1921_3_13__61_0.pdf Exposé géométrique d'un mémoire de Cayley sur les polygones de Poncelet*] , 1921

* [*http://archive.numdam.org/article/BSMF_1921__49__109_0.pdf Sur les diamètres rectilignes des courbes algébriques planes*] , 1921

* [*http://archive.numdam.org/article/AFST_1922_3_14__153_0.pdf Sur la théorie de la résiduation de Sylvester*] , 1922

* [*http://archive.numdam.org/article/BSMF_1924__52__315_1.pdf Remarques sur les deux premières démonstrations du théorème d'Euler relatif aux polyèdres*] , 1924

* [*http://archive.numdam.org/article/BSMF_1935__63__121_0.pdf Démonstration du théorème fondamental de la théorie projective des coniques faite à l'aide des droites focales de M. P. Robert*] , 1935

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**Henri Lebesgue**— Henri Léon Lebesgue Pour les articles homonymes, voir Lebesgue. Henri Léon Lebesgue Henri Lebesgue … Wikipédia en Français**Henri Lebesgue**— Henri Léon Lebesgue Henri Léon Lebesgue Henri Léon Lebesgue [ɑ̃ʁiː leɔ̃ ləˈbɛg] (* 28. Juni 1875 in … Deutsch Wikipedia**Henri-Leon Lebesgue**— Henri Léon Lebesgue Pour les articles homonymes, voir Lebesgue. Henri Léon Lebesgue Henri Lebesgue … Wikipédia en Français**Henri Leon Lebesgue**— Henri Léon Lebesgue Pour les articles homonymes, voir Lebesgue. Henri Léon Lebesgue Henri Lebesgue … Wikipédia en Français**Henri Léon Lebesgue**— Pour les articles homonymes, voir Lebesgue. Henri Léon Lebesgue Henri Lebesgue … Wikipédia en Français**Henri Leon Lebesgue**— Henri Léon Lebesgue Henri Léon Lebesgue Henri Léon Lebesgue [ɑ̃ʁiː leɔ̃ ləˈbɛg] (* 28. Juni 1875 in … Deutsch Wikipedia**Henri Léon Lebesgue**— Henri Léon Lebesgue [ … Deutsch Wikipedia**Henri Léon Lebesgue**— Henri Lebesgue. Nacimiento 28 de junio de 1875 … Wikipedia Español**LEBESGUE (H.)**— Le mathématicien Henri Lebesgue est l’un des fondateurs de l’analyse moderne. Presque tous ses travaux se rattachent à la théorie des fonctions de variables réelles. Sa conception de l’intégration et de la mesure renouvelle l’étude des problèmes… … Encyclopédie Universelle**Lebesgue covering dimension**— or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space. Contents 1 Definition 2 Examples 3 Properties 4 … Wikipedia