Henri Léon Lebesgue ForMemRS[1] (French: [ɑ̃ʁi leɔ̃
ləbɛɡ]; June 28, 1875 – July 26, 1941) was a French mathematician
most famous for his theory of integration, which was a generalization
of the 17th century concept of integration—summing the area between
an axis and the curve of a function defined for that axis. His theory
was published originally in his dissertation Intégrale, longueur,
aire ("Integral, length, area") at the
**University of Nancy**

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

Contents

1 Personal life
2 Mathematical career
3 Lebesgue's theory of integration
4 See also
5 References
6 External links

Personal life[edit]
**Henri Lebesgue**

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**

Beauvais and then at
**Lycée Saint-Louis**

Lycée Saint-Louis and Lycée
Louis-le-Grand in Paris.[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. 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
**Ph.D.**

Ph.D. 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 University of Rennes, lecturing there until 1906, when he moved
to the Faculty of Sciences of the University of Poitiers. In 1910
Lebesgue moved to the Sorbonne as a maître de conférences, being
promoted to professor starting with 1919. 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.[7] In 1922 he was
elected a member of the Académie des Sciences.
**Henri Lebesgue**

Henri Lebesgue died on
26 July 1941 in Paris.[6]
Mathematical career[edit]

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 Weierstrass' theorem on
approximation to continuous functions by polynomials. Between March
1899 and April 1901 Lebesgue published six notes in Comptes Rendus.
The first of these, unrelated to his development of Lebesgue
integration, dealt with the extension of
**Baire's theorem** to functions
of two variables. The next five dealt with surfaces applicable to a
plane, the area of skew 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 "Borel tract"
Leçons sur l'intégration et la recherche des fonctions primitives.
The problem of integration regarded as the search for a primitive
function is the keynote of the book. Lebesgue presents the problem of
integration in its historical context, addressing Augustin-Louis
Cauchy, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann. Lebesgue
presents six conditions which 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)."
Lebesgue shows that his conditions lead to the theory of measure and
measurable functions and the analytical and geometrical definitions of
the integral.
He turned next to trigonometric functions with his 1903 paper "Sur les
séries trigonométriques". He presented three major theorems in this
work: 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**

Fourier series is integrable
term by term. In 1904-1905 Lebesgue lectured once again at the
Collège de France, this time on trigonometrical series and he went on
to publish his lectures in another of the "Borel tracts". In this
tract he once again treats the subject in its historical context. He
expounds on Fourier series, Cantor-Riemann theory, the Poisson
integral and the Dirichlet problem.
In a 1910 paper, "Représentation trigonométrique approchée des
fonctions satisfaisant a une condition de Lipschitz" deals with the
**Fourier series**

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**

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[edit]

Approximation of the
**Riemann integral**

Riemann integral by rectangular areas.

Method of Lebesgue's integration.

This is a non-technical treatment from a historical point of view; see
the article
**Lebesgue integration**

Lebesgue integration for a technical treatment from a
mathematical point of view.

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**

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**

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. This surprising
relationship between two major geometric operations in calculus,
differentiation and integration, is now known as the Fundamental
Theorem of Calculus. It has allowed mathematicians to calculate a
broad class of integrals for the first time. 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,
**Augustin Cauchy**

Augustin Cauchy developed epsilon-delta limits,
and
**Bernhard Riemann**

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. As such, 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
domain of the function, Lebesgue looked at the codomain of the
function for his fundamental unit of area. Lebesgue's idea was to
first define measure, for both sets and functions on those sets. He
then proceeded to build the integral for what he called simple
functions; measurable functions that take only finitely many values.
Then he defined it for more complicated functions as the least upper
bound of all the integrals of simple functions smaller than the
function in question.
**Lebesgue integration**

Lebesgue integration has the property that every function defined over
a bounded interval with a
**Riemann integral**

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 into an integral
generalises easily to many other situations, leading to the modern
field of measure theory.
The Lebesgue integral is deficient in one respect. The Riemann
integral generalises to the improper
**Riemann integral**

Riemann integral to measure
functions whose domain of definition is not a closed interval. The
Lebesgue integral integrates many of these functions (always
reproducing the same answer when it did), but not all of them. For
functions on the real line, the
**Henstock integral** is an even more
general notion of integral (based on Riemann's theory rather than
Lebesgue's) that subsumes both
**Lebesgue integration**

Lebesgue integration and improper
Riemann integration. However, the
**Henstock integral** depends on
specific ordering features of 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[edit]

Dominated convergence theorem
Lebesgue covering dimension
Lebesgue point
Lebesgue's number lemma
Lebesgue spine
Lebesgue constant (interpolation)

References[edit]

^ a b c Burkill, J. C. (1944). "Henri Lebesgue. 1875-1941". Obituary
Notices of Fellows of the Royal Society. 4 (13): 483.
doi:10.1098/rsbm.1944.0001. JSTOR 768841.
^ "Prizes Awarded by the
**Paris**

Paris Academy of Sciences for 1914". Nature.
94 (2358): 518–519. 7 January 1915. doi:10.1038/094518a0.
^
**Henri Lebesgue**

Henri Lebesgue at the
**Mathematics**

Mathematics Genealogy Project
^ O'Connor, John J.; Robertson, Edmund F., "Henri Lebesgue", MacTutor
History of
**Mathematics**

Mathematics archive, University of St Andrews .
^ Hawking, Stephen W. (2005). God created the integers: the
mathematical breakthroughs that changed history. Running Press.
pp. 1041–87. ISBN 978-0-7624-1922-7.
^ a b McElroy, Tucker (2005). A to Z of mathematicians. Infobase
Publishing. p. 164. ISBN 978-0-8160-5338-4.
^ Perrin, Louis (2004). "Henri Lebesgue: Renewer of Modern Analysis".
In Le Lionnais, François. Great Currents of Mathematical Thought. 1
(2nd ed.). Courier Dover Publications.
ISBN 978-0-486-49578-1.
^ Pesin, Ivan N. (2014). Birnbaum, Z. W.; Lukacs, E., eds. Classical
and Modern Integration Theories. Academic Press. p. 94. 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'Ecole Supérieure 35 (1918), 36 (1919), 37
(1920)
^ Lebesgue, Henri (1918). "Remarques sur les théories de la mesure et
de l'intégration" (PDF). Annales de l'Ecole Supérieure. 35:
191–250.
^ Borel, Émile (1919). "L'intégration des fonctions non bornées"
(PDF). Annales de l'Ecole Supérieure. 36: 71–92.
^ Lebesgue, Henri (1920). "Sur une définition due à M. Borel (lettre
à M. le Directeur des Annales Scientifiques de l'École Normale
Supérieure)" (PDF). Annales de l'Ecole Supérieure. 37:
255–257.

External links[edit]

Henri Léon Lebesgue (28 juin 1875 [Rennes] - 26 juillet 1941 [Paris])
(in French)

Authority control

WorldCat Identities
VIAF: 7392947
LCCN: n50042567
ISNI: 0000 0001 0866 4934
GND: 118779117
SELIBR: 237550
SUDOC: 026975726
BNF: cb119118653 (data)
BIBSYS: 90830906
MGP: 86693
NLA: 35322256
NDL: 00447139
NKC: jx20050615012
ICCU: ITI