Henri Léon Lebesgue
(; June 28, 1875 – July 26, 1941) was a French
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known 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 during 1902.
Personal life
Henri Lebesgue was born on 28 June 1875 in
Beauvais,
Oise
Oise ( ; ; pcd, Oése) is a department in the north of France. It is named after the river Oise. Inhabitants of the department are called ''Oisiens'' () or ''Isariens'', after the Latin name for the river, Isara. It had a population of 829,419 ...
. Lebesgue's father was a
typesetter and his mother was a school
teacher
A teacher, also called a schoolteacher or formally an educator, is a person who helps students to acquire knowledge, competence, or virtue, via the practice of teaching.
''Informally'' the role of teacher may be taken on by anyone (e.g. w ...
. His parents assembled at home a library that the young Henri was able to use. His father died of
tuberculosis
Tuberculosis (TB) is an infectious disease usually caused by ''Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in w ...
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
The lycée Saint-Louis is a highly selective post-secondary school located in the 6th arrondissement of Paris, in the Latin Quarter. It is the only public French lycée exclusively dedicated to providing '' classes préparatoires aux grandes ...
and
Lycée Louis-le-Grand
The Lycée Louis-le-Grand (), also referred to simply as Louis-le-Grand or by its acronym LLG, is a public Lycée (French secondary school, also known as sixth form college) located on rue Saint-Jacques in central Paris. It was founded in the ...
in
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. ...
.
In 1894 Lebesgue was accepted at the
École Normale Supérieure
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education
Secondary education or post-primary education covers two phases on the International Standard Classification of Education sca ...
, 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
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated ...
'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 PHD or PhD may refer to:
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* ''Piled Higher and Deeper'', a web comic
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** Ph.D. (Ph.D. albu ...
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.
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
The University of Poitiers (UP; french: Université de Poitiers) is a public university located in Poitiers, France. It is a member of the Coimbra Group. It is multidisciplinary and contributes to making Poitiers the city with the highest stud ...
. 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
The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris ...
, where he lectured and did research for the rest of his life.
In 1922 he was elected a member of the
Académie des Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the ...
. Henri Lebesgue died on 26 July 1941 in
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. ...
.
[
]
Mathematical career
Lebesgue's first paper was published in 1898 and was titled "Sur l'approximation des fonctions". It dealt with Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
's 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 integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, on ...
s 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.
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
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
, Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
, 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 function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is i ...
s 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
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
is integrable term by term. In 1904-1905 Lebesgue lectured once again at the Collège de France
The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris ...
, 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
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriva ...
and the Dirichlet problem.
In a 1910 paper, "Représentation trigonométrique approchée des fonctions satisfaisant a une condition de Lipschitz" deals with the 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
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
and topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
. He also had a disagreement with Émile Borel about whose integral was more general. However, these minor forays pale in comparison to his contributions to real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include con ...
; 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
Integration
Integration may refer to:
Biology
* Multisensory integration
* Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technolo ...
is a mathematical operation that corresponds to the informal idea of finding the area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...
under the graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of a function. The first theory of integration was developed by Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scienti ...
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
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...
and Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
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
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
. 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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, mathematicians felt that Newton's and Leibniz's integral calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
did not have a rigorous foundation.
In the 19th century, Augustin Cauchy developed epsilon-delta limits, and Bernhard Riemann followed up on this by formalizing what is now called the Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of ...
. To define this integral, one fills the area under the graph with smaller and smaller rectangles and takes the limit of the sums
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fu ...
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
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
of the function, Lebesgue looked at the codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of all the integrals of simple functions smaller than the function in question.
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
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
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 to measure functions whose domain of definition is not a closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
.
The Lebesgue integral integrates many of these functions (always reproducing the same answer when it does), but not all of them.
For functions on the real line, the Henstock integral
Ralph Henstock (2 June 1923 – 17 January 2007) was an English mathematician and author. As an Integration theorist, he is notable for Henstock–Kurzweil integral. Henstock brought the theory to a highly developed stage without ever having encou ...
is an even more general notion of integral (based on Riemann's theory rather than Lebesgue's) that subsumes both Lebesgue integration and improper Riemann integration.
However, the Henstock integral depends on specific ordering features of the real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
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
* Lebesgue covering dimension
* Lebesgue's constants
* Lebesgue's decomposition theorem
* Lebesgue's density theorem
* Lebesgue differentiation theorem
* Lebesgue integration
* Lebesgue's lemma
* Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
* Lebesgue's number lemma
In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states:
:If the metric space (X, d) is compact and an open cover of X is given, then there exists a number \delta > 0 such ...
* Lebesgue point In mathematics, given a locally Lebesgue integrable function f on \mathbb^k, a point x in the domain of f is a Lebesgue point if
:\lim_\frac\int_ \!, f(y)-f(x), \,\mathrmy=0.
Here, B(x,r) is a ball centered at x with radius r > 0, and \lambda (B(x ...
* Lebesgue space
* Lebesgue spine In mathematics, in the area of potential theory, a Lebesgue spine or Lebesgue thorn is a type of set used for discussing solutions to the Dirichlet problem and related problems of potential theory. The Lebesgue spine was introduced in 1912 by Henri ...
* Lebesgue's universal covering problem
Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by definition is the least upper bound of the distance ...
* Lebesgue–Rokhlin probability space
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin i ...
* Lebesgue–Stieltjes integration
* Lebesgue–Vitali theorem
* Blaschke–Lebesgue theorem
* Borel–Lebesgue theorem
* Fatou–Lebesgue theorem
In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit su ...
* Riemann–Lebesgue lemma
* Walsh–Lebesgue theorem
* Dominated convergence theorem
* Osgood curve
In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover a convex set, distinguishing them from space-filling curves. Osgood curves are named ...
* Tietze extension theorem
* List of things named after Henri Lebesgue
{{unref, date=March 2022
The following are named after Henri Lebesgue:
*Blaschke–Lebesgue theorem
* Cantor–Lebesgue function
* Borel–Lebesgue theorem
*Fatou–Lebesgue theorem
* Lebesgue constant
* Lebesgue covering dimension
*Lebesgue diff ...
References
External links
*
Henri Léon Lebesgue (28 juin 1875 [Rennes
- 26 juillet 1941 [Paris">ennes">Henri Léon Lebesgue (28 juin 1875 [Rennes
- 26 juillet 1941 [Paris]
{{DEFAULTSORT:Lebesgue, Henri
1875 births
1941 deaths
People from Beauvais
20th-century French mathematicians
Measure theorists
Functional analysts
Mathematical analysts
Intuitionism
École Normale Supérieure alumni
Lycée Louis-le-Grand alumni
Members of the French Academy of Sciences
Foreign Members of the Royal Society
University of Poitiers faculty