Henri Lebesgue
Henri Lebesgue | |
---|---|
École Normale Supérieure University of Paris | |
Known for | Lebesgue integration Lebesgue measure |
Awards | Fellow of the Royal Society[1] Poncelet Prize for 1914[2] |
Scientific career | |
Fields | Mathematics |
Institutions | University of Rennes University of Poitiers University of Paris Collège de France |
Doctoral advisor | Émile Borel |
Doctoral students | Paul Montel Zygmunt Janiszewski Georges de Rham |
Henri Léon Lebesgue
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
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
Mathematical career
Lebesgue's first paper was published in 1898 and was titled "Sur l'approximation des fonctions". It dealt with
His lectures from 1902 to 1903 were collected into a "
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
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
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
In the 19th century,
Lebesgue invented a new method of integration to solve this problem. Instead of using the areas of rectangles, which put the focus on the
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
The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the
See also
- Lebesgue covering dimension
- Lebesgue constants
- Lebesgue's decomposition theorem
- Lebesgue's density theorem
- Lebesgue differentiation theorem
- Lebesgue integration
- Lebesgue's lemma
- Lebesgue measure
- Lebesgue's number lemma
- Lebesgue point
- Lebesgue space
- Lebesgue spine
- Lebesgue's universal covering problem
- Lebesgue–Rokhlin probability space
- Lebesgue–Stieltjes integration
- Lebesgue–Vitali theorem
- Blaschke–Lebesgue theorem
- Borel–Lebesgue theorem
- Fatou–Lebesgue theorem
- Riemann–Lebesgue lemma
- Walsh–Lebesgue theorem
- Dominated convergence theorem
- Osgood curve
- Tietze extension theorem
- List of things named after Henri Lebesgue
References
- ^ S2CID 122854745.
- doi:10.1038/094518a0.
- ^ Henri Lebesgue at the Mathematics Genealogy Project
- ^ O'Connor, John J.; Robertson, Edmund F., "Henri Lebesgue", MacTutor History of Mathematics Archive, University of St Andrews
- ISBN 978-0-7624-1922-7.
- ^ ISBN 978-0-8160-5338-4.
- ISBN 978-0-486-49578-1.
- ISBN 9781483268699.
Borel's assertion that his integral was more general compared to Lebesgue's integral was the cause of the dispute between Borel and Lebesgue in the pages of Annales de l'École Supérieure 35 (1918), 36 (1919), 37 (1920)
- (PDF) from the original on 2009-09-16.
- (PDF) from the original on 2014-08-05.
- (PDF) from the original on 2009-09-16.
External links
- Media related to Henri-Léon Lebesgue at Wikimedia Commons
- Henri Léon Lebesgue (28 juin 1875 [Rennes] - 26 juillet 1941 [Paris]) (in French)