Henri Lebesgue

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Henri Lebesgue
École Normale Supérieure
University of Paris
Known forLebesgue integration
Lebesgue measure
AwardsFellow of the Royal Society[1]
Poncelet Prize for 1914[2]
Scientific career
FieldsMathematics
InstitutionsUniversity of Rennes
University of Poitiers
University of Paris
Collège de France
Doctoral advisorÉmile Borel
Doctoral studentsPaul Montel
Zygmunt Janiszewski
Georges de Rham

Henri Léon Lebesgue

University of Nancy during 1902.[3][4]

Personal life

Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise. Lebesgue's father was a typesetter and his mother was a school teacher. His parents assembled at home a library that the young Henri was able to use. His father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. As he showed a remarkable talent for mathematics in primary school, one of his instructors arranged for community support to continue his education at the Collège de Beauvais and then at Lycée Saint-Louis and Lycée Louis-le-Grand in Paris.[5]

In 1894, Lebesgue was accepted at the

PhD from the Sorbonne with the seminal thesis on "Integral, Length, Area", submitted with Borel, four years older, as advisor.[6]

Lebesgue married the sister of one of his fellow students, and he and his wife had two children, Suzanne and Jacques.

After publishing his thesis, Lebesgue was offered in 1902 a position at the

Académie des Sciences. Henri Lebesgue died on 26 July 1941 in Paris.[6]

Mathematical career

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

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 "

and the analytical and geometrical definitions of the integral.

He turned next to

.

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 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 and topology. He also had a disagreement with Émile Borel about whose integral was more general.[8][9][10][11] 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.

Lebesgue's theory of integration

Approximation of the Riemann integral by rectangular areas

integral calculus
did not have a rigorous foundation.

In the 19th century,

Augustin Cauchy developed epsilon-delta limits, and Bernhard Riemann followed up on this by formalizing 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. Thus, 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

least upper bound
of all the integrals of simple functions smaller than the function in question.

Lebesgue integration has the property that every function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree. Furthermore, every bounded function on a closed bounded interval has a Lebesgue integral and 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 measure, which extends the idea of length from 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 theory
.

The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the

real 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.

See also

References

External links