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Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, République française), is a transcontinental country This is a list of co ...

French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

mathematician
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 A university ( la, universitas, 'a whole') is an institution Institutions, according to Samuel P. Huntington, are "stable, valued, recurring patterns of behavior". Institutions can refer to mechanisms which govern the behavior Behavio ...
during 1902.


Personal life

Henri Lebesgue was born on 28 June 1875 in
Beauvais Beauvais ( , ; historical English: Beawayes, Beeway, Boway; pcd, Bieuvais) is a city and commune in northern France. It serves as the capital of the Oise Departments of France, département, in the Hauts-de-France Regions of France, region. Bea ...

Beauvais
,
Oise Oise ( ; ; pcd, Oése) is a department Department may refer to: * Departmentalization, division of a larger organization into parts with specific responsibility Government and military *Department (country subdivision), a geographical and adm ...
. Lebesgue's father was a
typesetter on a composing stick on a type case. , letter founder, from the 1728 edition of '' Cyclopaedia, or an Universal Dictionary of Arts and Sciences, Cyclopaedia''. . Typesetting is the composition of Written language, text by means of arranging ph ...
and his mother was a school
teacher A teacher, also called a schoolteacher or formally an educator, is a person who helps student A student is primarily a person enrolled in a school A school is an educational institution designed to provide learning spaces and le ...

teacher
. 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 An infection is the invasion of an organism's body Tissue (biology), tissues by Pathogen, disease-causing agents, their multiplication, and the reaction of host (biology), host tissues to the in ...

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 The lycée Saint-Louis is a post-secondary school located in the 6th arrondissement The 6th arrondissement of Paris (''VIe arrondissement'') is one of the 20 Arrondissements of Paris, arrondissements of the capital city of France. In spoken Fre ...

Lycée Saint-Louis
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 In France France (), officially the French Republic (french: link=no, République française), is a country primarily l ...

Lycée Louis-le-Grand
in
Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,175,601 residents , in an area of more than . Since the 17th century, Paris ha ...

Paris
. In 1894 Lebesgue was accepted at the
École Normale Supérieure École may refer to: * an elementary school in the French educational stages Educational stages are subdivisions of formal learning, typically covering early childhood education, primary education, secondary education and tertiary education. ...
, 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 The Sorbonne ( , , ) is a building in the Latin Quarter The Latin Quarter of Paris (french: Quartier latin, ) is an area in the 5th and the 6th arrondissements of Paris The city of Paris is divided into twenty ''municipal arrondissem ...
, where he learned about
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such top ...
's work on the incipient
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
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 a ...
's work on the
Jordan measure In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (Arc length, length, area (mathematics), area, volume) to shapes more complicated than, for example, a triangle, Disk (mathematics), ...
. In 1899 he moved to a teaching position at the Lycée Central in
Nancy Nancy may refer to: Places France * Nancy, France, a city in the northeastern French department of Meurthe-et-Moselle and formerly the capital of the duchy of Lorraine ** Arrondissement of Nancy, surrounding and including the city of Nancy * ...

Nancy
, while continuing work on his doctorate. In 1902 he earned his
Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'' ...
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 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 (Law, Arts and Science) in 1885. In 1969, it was divided into two new universities: *University of Ren ...
, 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 #REDIRECT Public university #REDIRECT Public university #REDIRECT Public university#REDIRECT Public university A public university or public college is a u ...
. 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, i ...
, 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 A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discip ...
. Henri Lebesgue died on 26 July 1941 in
Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,175,601 residents , in an area of more than . Since the 17th century, Paris ha ...

Paris
.


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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

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 The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
to functions of two variables. The next five dealt with surfaces applicable to a plane, the area of skew
polygons In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

polygons
,
surface integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
). 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

Augustin-Louis Cauchy
,
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For c ...

Peter Gustav Lejeune Dirichlet
, and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and the analytical and geometrical definitions of the integral. He turned next to
trigonometric Trigonometry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

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 In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an Lp space, ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and ...
), and that a
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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, i ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in whic ...
and the
Dirichlet problem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. 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 In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...
, with an evaluation of the order of magnitude of the remainder term. He also proves that the
Riemann–Lebesgue lemma In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an Lp space, ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and ...
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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
and
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
. He also had a disagreement with
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such top ...
about whose integral was more general. However, these minor forays pale in comparison to his contributions to
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

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

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 technology, ...

Integration
is a mathematical operation that corresponds to the informal idea of finding the
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
. The first theory of integration was developed by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

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 Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...
discovered the idea that integration was intrinsically linked to
differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in marketing * Differentiated service, a service that varies with the identity o ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
. 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 Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, mathematicians felt that Newton's and Leibniz's
integral calculus In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
did not have a rigorous foundation. In the 19th century,
Augustin 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. He was ...

Augustin Cauchy
developed epsilon-delta
limits Limit or Limits may refer to: Arts and media * Limit (manga), ''Limit'' (manga), a manga by Keiko Suenobu * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (song), "Limits" (song), ...

limits
, and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
followed up on this by formalizing what is now called the
Riemann integral In the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
. To define this integral, one fills the area under the graph with smaller and smaller
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
s and takes the limit of the
sums In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
of the function, Lebesgue looked at the
codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 function In the mathematics, mathematical field of real analysis, a simple function is a real number, real (or complex number, complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that ...
s; measurable functions that take only finitely many values. Then he defined it for more complicated functions as the
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

least upper bound
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 Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measu ...
, which extends the idea of
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
from intervals to a very large class of sets, called measurable sets (so, more precisely,
simple function In the mathematics, mathematical field of real analysis, a simple function is a real number, real (or complex number, complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that ...
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 may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measu ...
into an integral generalises easily to many other situations, leading to the modern field of
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
and so does not generalise to allow integration in more general spaces (say,
manifolds In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
), while the Lebesgue integral extends to such spaces quite naturally.


See also

*
Lebesgue covering dimension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
* Lebesgue's constants *
Lebesgue's decomposition theorem In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two sigma-finite measure, σ-finite signed measures \mu and \nu on a measurable space (\Omega,\Sigma), there exist two σ-finite signed measure ...
*
Lebesgue's density theorem In mathematics, Lebesgue's density theorem states that for any Lebesgue measure, Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost everywhere, almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at a ...
*
Lebesgue differentiation theoremIn mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for H ...
*
Lebesgue integration In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the -axis. The Lebesgue integral, ...
* Lebesgue's lemma *
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
* Lebesgue's number lemma * Lebesgue point * Lebesgue space * Lebesgue spine * Lebesgue's universal covering problem * Lebesgue–Rokhlin probability space *
Lebesgue–Stieltjes integration In measure-theoretic analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material wi ...
* *
Blaschke–Lebesgue theorem In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all Curve of constant width, curves of given constant width. In the form that every curve of a given width has area at least as large as the ...
* Borel–Lebesgue theorem *
Fatou–Lebesgue theorem In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequality (mathematics), inequalities relating the integrals (in the sense of Lebesgue integration, Lebesgue) of the limit superior and limit inferior, limit inferior and the limi ...
*
Riemann–Lebesgue lemma In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an Lp space, ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and ...
* Walsh–Lebesgue theorem *
Dominated convergence theorem In measure theory, Henri Lebesgue, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence (mathematics), convergence of a sequence of Function (mathematics), functions implies convergence in ...
*
Osgood curve In mathematics, an Osgood curve is a non-self-intersecting curve (either a Jordan curve or a Jordan_curve_theorem#Definitions_and_the_statement_of_the_Jordan_theorem, Jordan arc) of positive area. More formally, these are curves in the Euclidean ...

Osgood curve
*
Tietze extension theorem In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...


References


External links

*
Henri_Léon_Lebesgue_(28_juin_1875_[Rennes
-_26_juillet_1941_[Paris.html" ;"title="ennes">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