- Henri Lebesgue
name =Henri Lebesgue
death_date =death date and age|1941|7|26|1875|6|28
École Normale Supérieure, France
Paul Montel Zygmunt Janiszewski Georges de Rham
Lebesgue integration Lebesgue measure
Henri Léon Lebesgue IPA| [ɑ̃ʁiː leɔ̃ ləˈbɛg] (
June 28, 1875, Beauvais– July 26, 1941, Paris) was a French mathematician, 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 of Nancyin 1902.
Lebesgue's father was a typesetter, who died of
tuberculosiswhen 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.
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 Theoremto 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 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 (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 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, 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 seriesrepresenting a bounded function is a Fourier series, that the nth Fourier coefficient tends to zero (the
Riemann-Lebesgue lemma), and that a Fourier seriesis 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 integraland 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 lemmais 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 integralgeneralizes 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 analysisand 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 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 integrationfor a technical treatment from a mathematical point of view.
Integration is a mathematical operation that corresponds to the informal idea of finding the
areaunder the graph of a function. The first theory of integration was developed by Archimedesin 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 Newtonand 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 calculusdid not have a rigorous foundation.
In the nineteenth century,
Augustin Cauchyfinally developed a rigorous theory of limits, and Bernhard Riemannfollowed up on this by formalising 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
codomainof 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 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 of lengthfrom 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 integralto 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 integralis 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 lineand so does not generalise to allow integration in more general spaces (say, manifolds), while the Lebesgue integral extends to such spaces quite naturally.
Dominated convergence theorem
Lebesgue covering dimension
Lebesgue's number lemma
Lebesgue constant (interpolation)
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]
Henri 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
Wikimedia Foundation. 2010.
Look at other dictionaries:
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