**Lebesgue** formulated the theory of measure in 1901 and the following year he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral.

- Henri began his studies at the Collège de Beauvais, then he went to Paris where he studied first at the Lycée Saint Louis and then at the Lycée Louis-le-Grand.
- Lebesgue entered the École Normale Supérieure in Paris in 1894 and was awarded his teaching diploma in mathematics in 1897.
- Later there would be considerable rivalry between Baire and Lebesgue which we refer to below.
- Building on the work of others, including that of Émile Borel and Camille Jordan, Lebesgue formulated the theory of measure in 1901 and in his famous paper "Sur une généralisation de l'intégrale définie" Ⓣ(On a generalization of the definite integral), which appeared in the Comptes Rendus on 29 April 1901, he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions.
- This outstanding piece of work appears in Lebesgue's doctoral dissertation, Intégrale, longueur, aire Ⓣ(Integral, length, area) , presented to the Faculty of Science in Paris in 1902, and the 130 page work was published in Milan in the Annali di Matematica in the same year.
- Having graduated with his doctorate, Lebesgue obtained his first university appointment when in 1902 he became mâitre de conférences in mathematics at the Faculty of Science in Rennes.
- One honour which Lebesgue received at an early stage in his career was an invitation to give the Cours Peccot at the Collège de France.
- Lebesgue first fell out with Baire in 1904, when Baire gave the Cours Peccot at the Collège de France, over who had the most right to teach such a course.
- Lebesgue wrote two monographs "Leçons sur l'intégration et la recherche des fonctions primitives" Ⓣ(Lectures on integration and research on primitive functions) (1904) and Leçons sur les séries trigonométriques Ⓣ(Lectures on trigonometric series) (1906) which arose from these two lecture courses and served to make his important ideas more widely known.
- Let us attempt to indicate the way that the Lebesgue integral enabled many of the problems associated with integration to be solved.
- In 1905 Lebesgue gave a deep discussion of the various conditions Lipschitz and Jordan had used in order to ensure that a function f(x)f (x)f(x) is the sum of its Fourier series.
- What Lebesgue was able to show was that term by term integration of a uniformly bounded series of Lebesgue integrable functions was always valid.
- What made the new definition important was that Lebesgue was able to recognise in it an analytic tool capable of dealing with - and to a large extent overcoming - the numerous theoretical difficulties that had arisen in connection with Riemann's theory of integration.
- The problems posed by these difficulties motivated all of Lebesgue's major results.
- Lebesgue held his post at the Sorbonne until 1918 when he was promoted to Professor of the Application of Geometry to Analysis.
- It is interesting that Lebesgue did not concentrate throughout his career on the field which he had himself started.
- This was because his work was a striking generalisation, yet Lebesgue himself was fearful of generalisations.
- By 1922 when he published Notice sur les travaux scientifique de M Henri Lebesgue he had written nearly 90 books and papers.
- This ninety-two page work also provides an analysis of the contents of Lebesgue's papers.
- Lebesgue was honoured with election to many academies.

Born 28 June 1875, Beauvais, Oise, Picardie, France. Died 26 July 1941, Paris, France.

View full biography at MacTutor

Analysis, Geometry, Set Theory

Definitions: 1

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive