His theory was published originally in his dissertation Intégrale, longueur, aire ("Integral, length, area") at the University of Nancy during 1902.

[5] In 1894, Lebesgue was accepted at the École Normale Supérieure, where he continued to focus his energy on the study of mathematics, graduating in 1897.

In 1902 he earned his PhD from the Sorbonne with the seminal thesis on "Integral, Length, Area", submitted with Borel, four years older, as advisor.

In 1921 he left the Sorbonne to become professor of mathematics at the Collège de France, where he lectured and did research for the rest of his life.

Lebesgue's great thesis, Intégrale, longueur, aire, with the full account of this work, appeared in the Annali di Matematica in 1902.

[1] His lectures from 1902 to 1903 were collected into a "Borel tract" Leçons sur l'intégration et la recherche des fonctions primitives.

Lebesgue presents the problem of integration in its historical context, addressing Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann.

Lebesgue once wrote, "Réduites à des théories générales, les mathématiques seraient une belle forme sans contenu."

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 by Archimedes in the 3rd 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 17th century, Isaac Newton and Gottfried Wilhelm Leibniz discovered the idea that integration was intrinsically linked to differentiation, the latter being a way of measuring how quickly a function changed at any given point on the graph.

However, unlike Archimedes' method, which was based on Euclidean geometry, mathematicians felt that Newton's and Leibniz's integral calculus did not have a rigorous foundation.

In the 19th century, Karl Weierstrass developed the rigorous epsilon-delta definition of a limit, which is still accepted and used by mathematicians today.

He built on previous but non-rigorous work by Augustin Cauchy, who had used the non-standard notion of infinitesimally small numbers, today rejected in standard mathematical analysis.

In 1947 Norbert Wiener claimed that the Lebesgue integral had unexpected but important implications in establishing the validity of Willard Gibbs' work on the foundations of statistical mechanics.

[12] The notions of average and measure were urgently needed to provide a rigorous proof of Gibbs' ergodic hypothesis.