Henri Lebesgue

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Lore & Background

Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise. His 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. 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. After graduation he remained at the École Normale Supérieure for two years, working in the library, where he became aware of the research on discontinuity done at that time by René-Louis Baire, a recent graduate of the school. At the same time he started his graduate studies at the Sorbonne, where he learned about Émile Borel's work on the incipient measure theory and Camille Jordan's work on the Jordan measure. In 1899 he moved to a teaching position at the Lycée Central in Nancy, while continuing work on his doctorate. 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. Lebesgue married the sister of one of his fellow students, and he and his wife had two children, Suzanne and Jacques.

Reader's Guide

Lebesgue's theory of integration addressed limitations of the Riemann integral, which fails for some functions. 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. His 1902 dissertation developed the theory of measure in its first chapter, and in the second chapter defined the integral both geometrically and analytically. His lectures from 1902 to 1903 were collected into a "Borel tract" Leçons sur l'intégration et la recherche des fonctions primitives, where he presented six conditions 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)." He also made major contributions to trigonometric series, proving in his 1903 paper "Sur les séries trigonométriques" that a trigonometrical series representing a bounded function is a Fourier series, that the nth Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series is integrable term by term. 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 his career, Lebesgue also made forays into complex analysis and topology, and had a disagreement with Émile Borel about whose integral was more general.

Did You Know?

• Lebesgue's father was a typesetter and his mother was a school teacher; they assembled a home 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.
• Lebesgue's PhD thesis, Intégrale, longueur, aire, was submitted with Émile Borel, four years older, as advisor.
• 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.")
• Lebesgue had a disagreement with Émile Borel about whose integral was more general.