Henri Lebesgue
Encyclopedia
Henri Léon Lebesgue was a French
mathematician
most famous for his theory of integration
, which was a generalization of the seventeenth 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.
, 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. Unfortunately 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 instructor arranged for community support to continue his education at the Collège de Beauvais
and then at Lycée SaintLouis
and Lycée LouisleGrand
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 libarary, where he became aware of the research on discontinuity done at that time by RenéLouis Baire
, a recent graduate of the school. At the same time he started his graduate studies at the Sorbonne
, were 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 Centrale in Nancy
, while continuing work on his doctorate. In 1902 he earned his Ph.D.
from the Sorbonne with the seminal thesis on "Integral, Length, Area", submitted with the fouryear older Borel 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
. 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. In 1922 he was ellected a member of the Académie française
. Henri Lebesgue died on 26 July 1941 in Paris
.
. 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
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 Cauchy, Dirichlet, and Riemann. Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is "If the sequence f_{n}(x) increases to the limit f(x), the integral of f_{n}(x) tends to the integral of f(x)." Lebesgue shows that his conditions lead to the theory of measure and measurable function
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 n^{th} Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series
is integrable term by term. In 19041905 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, CantorRiemann 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 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 measuretheoretic 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 measuretheoretic 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 Borel (called teilweise heftig) with regards to effective calculation. 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.
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
in the third 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 seventeenth century, Isaac Newton
and Gottfried Wilhelm Leibniz
independently discovered the idea that integration was roughly the inverse operation of differentiation
, the latter being a way of measuring how quickly a function changed at any given point on the graph. This allowed mathematicians to calculate a broad class of integrals for the first time. However, unlike Archimedes' method, which was based on Euclidean geometry
, Newton's and Leibniz's integral calculus did not have a rigorous foundation.
In the nineteenth century, Augustin Cauchy finally developed a rigorous theory of limits
, and 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 rectangle
s and takes the limit of the sum
s 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 build the integral for what he called simple function
s, 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 has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.
But 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 function
s 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.
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 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.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semipresidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
most famous for his theory of integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
, which was a generalization of the seventeenth 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
Henry Lebesgue was born on 28 June 1875 in BeauvaisBeauvais
Beauvais is a city approximately by highway north of central Paris, in the northern French region of Picardie. It currently has a population of over 60,000 inhabitants. History :...
, Oise
Oise
Oise is a department in the north of France. It is named after the river Oise.History:Oise is one of the original 83 departments created during the French Revolution on March 4, 1790...
. Lebesgue's father was a typesetter
Typesetting
Typesetting is the composition of text by means of types.Typesetting requires the prior process of designing a font and storing it in some manner...
and his mother was a school teacher
Teacher
A teacher or schoolteacher is a person who provides education for pupils and students . The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional...
. His parents assembled at home a library that the young Henri was able to use. Unfortunately his father died of tuberculosis
Tuberculosis
Tuberculosis, MTB, or TB is a common, and in many cases lethal, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis. Tuberculosis usually attacks the lungs but can also affect other parts of the body...
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 instructor arranged for community support to continue his education at the Collège de Beauvais
Collège de Beauvais
The College of Beauvais was in Paris in what is now the Rue Jean de Beauvais. At the end of the 17th century and at the beginning of the 18th century, it was one of the leading schools of France, educating pupils whose parents were prominent in the French establishment.The college was founded in...
and then at Lycée SaintLouis
Lycée SaintLouis
The lycée SaintLouis is a higher education establishment located in the VIe arrondissement of Paris, in the Latin Quarter. It is the only public French lycée exclusively dedicated to classes préparatoires aux grandes écoles...
and Lycée LouisleGrand
Lycée LouisleGrand
The Lycée LouisleGrand is a public secondary school located in Paris, widely regarded as one of the most rigorous in France. Formerly known as the Collège de Clermont, it was named in king Louis XIV of France's honor after he visited the school and offered his patronage.It offers both a...
in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the ÎledeFrance region...
.
In 1894 Lebesgue was accepted at the École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
, 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 libarary, where he became aware of the research on discontinuity done at that time by RenéLouis Baire
RenéLouis Baire
RenéLouis Baire was a French mathematician most famous for his Baire category theorem, which helped to generalize and prove future theorems...
, a recent graduate of the school. At the same time he started his graduate studies at the Sorbonne
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
, were he learned about Émile Borel
Émile Borel
Félix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in SaintAffrique, Aveyron. Along with RenéLouis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...
's work on the incipient measure theory and Camille Jordan
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
's work on the Jordan measure
Jordan measure
In mathematics, the Peano–Jordan measure is an extension of the notion of size to shapes more complicated than, for example, a triangle, disk, or parallelepiped....
. In 1899 he moved to a teaching position at the Lycée Centrale in Nancy
Nancy
Nancy is a city in the northeastern French department of MeurtheetMoselle, and formerly the capital of the Duchy of Lorraine, and then the French province of the same name.The city is the head of the department...
, while continuing work on his doctorate. In 1902 he earned his Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
from the Sorbonne with the seminal thesis on "Integral, Length, Area", submitted with the fouryear older Borel 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
University of Rennes
The University of Rennes was a French university located in the city of Rennes. It was established by the union of the 3 faculties of the city in 1885. In 1969, it was divided in two new universities:* the University of Rennes 1...
, lecturing there until 1906, when he moved to the Faculty of Sciences of the University of Poitiers
University of Poitiers
The University of Poitiers is a university in Poitiers, France. It is a member of the Coimbra Group.History:Founded in 1431 by Pope Eugene IV and chartered by King Charles VII, the University of Poitiers was originally composed of five faculties: theology, canon law, civil law, medicine, and...
. 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
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue SaintJacques and Rue des Écoles...
, where he lectured and did research for the rest of his life. In 1922 he was ellected a member of the Académie française
Académie française
L'Académie française , also called the French Academy, is the preeminent French learned body on matters pertaining to the French language. The Académie was officially established in 1635 by Cardinal Richelieu, the chief minister to King Louis XIII. Suppressed in 1793 during the French Revolution,...
. Henri Lebesgue died on 26 July 1941 in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the ÎledeFrance region...
.
Mathematical career
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 RendusComptes rendus
Comptes rendus de l'Académie des Sciences, or simply Comptes rendus, is a French scientific journal which has been published since 1666. It is the proceedings of the French Academy of Sciences...
. 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
Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
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
Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by JosephLouis Lagrange in 1760. However, it is named after Joseph Plateau who was interested in soap films. The problem is considered part of the calculus of variations...
. 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
Émile Borel
Félix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in SaintAffrique, Aveyron. Along with RenéLouis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...
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 Cauchy, Dirichlet, and Riemann. Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is "If the sequence f_{n}(x) increases to the limit f(x), the integral of f_{n}(x) tends to the integral of f(x)." Lebesgue shows that his conditions lead to the theory of measure and measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structurepreserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
s and the analytical and geometrical definitions of the integral.
He turned next to trigonometric
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
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 n^{th} Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
is integrable term by term. In 19041905 Lebesgue lectured once again at the Collège de France
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue SaintJacques and Rue des Écoles...
, 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, CantorRiemann theory, the Poisson integral and the Dirichlet problem
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
.
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 measuretheoretic 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 measuretheoretic framework.
During the course of his career, Lebesgue also made forays into the realms of complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. He also had a disagreement with Borel (called teilweise heftig) with regards to effective calculation. However, these minor forays pale in comparison to his contributions to real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
; 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.
Lebesgue's theory of integration
This is a nontechnical treatment from a historical point of view; see the article Lebesgue integrationLebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
for a technical treatment from a mathematical point of view.
Integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
is a mathematical operation that corresponds to the informal idea of finding the area
Area
Area is a quantity that expresses the extent of a twodimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
under the graph of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. The first theory of integration was developed by Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
in the third century BC with his method of quadratures
Quadrature (mathematics)
Quadrature — historical mathematical term which means calculating of the area. Quadrature problems have served as one of the main sources of mathematical analysis. History :...
, but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the seventeenth century, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and Gottfried Wilhelm Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
independently discovered the idea that integration was roughly the inverse operation of differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, the latter being a way of measuring how quickly a function changed at any given point on the graph. This 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
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, Newton's and Leibniz's integral calculus did not have a rigorous foundation.
In the nineteenth century, Augustin Cauchy finally developed a rigorous theory of limits
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
, and Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
followed up on this by formalizing what is now called the Riemann integral
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. The Riemann integral is unsuitable for many theoretical purposes...
. To define this integral, one fills the area under the graph with smaller and smaller rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a nonsquare rectangle...
s and takes the limit of the sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUpManager* Software User’s Manual,as from DODSTD2 167A, and MILSTD498...
s 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
Codomain
In mathematics, the codomain or target set 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...
of the function for his fundamental unit of area.
Lebesgue's idea was to first build the integral for what he called simple function
Simple function
In the mathematical field of real analysis, a simple function is a realvalued function over a subset of the real line which attains only a finite number of values...
s, measurable functions that take only finitely many values.
Then he defined it for more complicated functions as the least upper bound
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of all the integrals of simple functions smaller than the function in question.
Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.
But 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
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ndimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
, which extends the idea of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
from intervals to a very large class of sets, called measurable sets (so, more precisely, simple function
Simple function
In the mathematical field of real analysis, a simple function is a realvalued function over a subset of the real line which attains only a finite number of values...
s 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
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
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.
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 and improper Riemann integration.
However, the Henstock integral depends on specific ordering features of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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
 Dominated convergence theoremDominated convergence theoremIn measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
 Lebesgue covering dimensionLebesgue covering dimensionLebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.Definition:...
 Lebesgue point
 Lebesgue's number lemmaLebesgue's number lemmaIn topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states:Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.References:External links:*...
 Lebesgue spineLebesgue spineIn 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...
 Lebesgue constant (interpolation)Lebesgue constant (interpolation)In mathematics, the Lebesgue constants give an idea of how good the interpolant of a function is in comparison with the best polynomial approximation of the function...
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Henri Léon Lebesgue (28 juin 1875 [Rennes  26 juillet 1941 [Paris])]