396:
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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
557:; 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.
55:
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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 Fourier coefficient tends to zero (the
721:
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
504:, 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
617:, 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
297:
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
321:, where he continued to focus his energy on the study of mathematics, graduating in 1897. After graduation he remained at the Ăcole Normale SupĂ©rieure for two years, working in the library, where he became aware of the research on
261:, which was a generalization of the 17th-century concept of integrationâsumming the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation
530:
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.")
667:
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
105:
456:. 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
656:
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. Thus, they have no
Riemann integral.
515:
In a 1910 paper, "Représentation trigonométrique approchée des fonctions satisfaisant a une condition de
Lipschitz" deals with the Fourier series of functions satisfying a
717:. 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
1365:
1117:
Borel's assertion that his integral was more general compared to
Lebesgue's integral was the cause of the dispute between Borel and Lebesgue in the pages of
1355:
395:
538:
generalizes
RiemannâStieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.
429:
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,
890:
621:. It has allowed mathematicians to calculate a broad class of integrals for the first time. However, unlike Archimedes' method, which was based on
1360:
1345:
1256:
1170:
659:
Lebesgue invented a new method of integration to solve this problem. Instead of using the areas of rectangles, which put the focus on the
433:, with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops the theory of measure (see
1320:
1213:
410:'s theorem on approximation to continuous functions by polynomials. Between March 1899 and April 1901 Lebesgue published six notes in
860:
698:
are functions that take a finite number of values, and each value is taken on a measurable set). Lebesgue's technique for turning a
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468:. Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is "If the sequence f
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1335:
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from the
Sorbonne with the seminal thesis on "Integral, Length, Area", submitted with Borel, four years older, as advisor.
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605:, but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the 17th century,
244:
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375:, being promoted to professor starting in 1919. In 1921 he left the Sorbonne to become professor of mathematics at the
1232:"Sur une dĂ©finition due Ă M. Borel (lettre Ă M. le Directeur des Annales Scientifiques de l'Ăcole Normale SupĂ©rieure)"
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Lebesgue married the sister of one of his fellow students, and he and his wife had two children, Suzanne and Jacques.
1110:
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437:). In the second chapter he defines the integral both geometrically and analytically. The next chapters expand the
618:
535:
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Lebesgue's first paper was published in 1898 and was titled "Sur l'approximation des fonctions". It dealt with
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about whose integral was more general. However, these minor forays pale in comparison to his contributions to
1315:
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to functions of two variables. The next five dealt with surfaces applicable to a plane, the area of skew
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The first of these, unrelated to his development of Lebesgue integration, dealt with the extension of
379:, where he lectured and did research for the rest of his life. In 1922 he was elected a member of the
710:
1070:
Perrin, Louis (2004). "Henri Lebesgue: Renewer of Modern Analysis". In Le Lionnais, François (ed.).
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into an integral generalises easily to many other situations, leading to the modern field of
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notes dealing with length, area and applicable surfaces. The final chapter deals mainly with
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1310:
1305:
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The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the
575:
445:. This dissertation is considered to be one of the finest ever written by a mathematician.
442:
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329:, a recent graduate of the school. At the same time he started his graduate studies at the
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from intervals to a very large class of sets, called measurable sets (so, more precisely,
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648:. To define this integral, one fills the area under the graph with smaller and smaller
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As part of the development of Lebesgue integration, Lebesgue invented the concept of
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476:(x) tends to the integral of f(x)." Lebesgue shows that his conditions lead to the
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is a mathematical operation that corresponds to the informal idea of finding the
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of all the integrals of simple functions smaller than the function in question.
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is a best possible result for continuous functions, and gives some treatment to
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During the course of his career, Lebesgue also made forays into the realms of
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God created the integers: the mathematical breakthroughs that changed history
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and so does not generalise to allow integration in more general spaces (say,
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500:
is integrable term by term. In 1904-1905 Lebesgue lectured once again at the
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After publishing his thesis, Lebesgue was offered in 1902 a position at the
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This is a historical overview. For a technical mathematical treatment, see
488:
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257:; June 28, 1875 â July 26, 1941) was a French mathematician known for his
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220:
176:
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1208:
1165:
730:), while the Lebesgue integral extends to such spaces quite naturally.
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534:
In measure-theoretic analysis and related branches of mathematics, the
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many values. Then he defined it for more complicated functions as the
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functions with a Lebesgue integral that have no Riemann integral.
349:. In 1899 he moved to a teaching position at the Lycée Central in
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discovered the idea that integration was intrinsically linked to
454:
Leçons sur l'intégration et la recherche des fonctions primitives
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Leçons sur l'integration et la recherche des fonctions primitives
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and the analytical and geometrical definitions of the integral.
353:, while continuing work on his doctorate. In 1902 he earned his
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1146:"Remarques sur les théories de la mesure et de l'intégration"
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Pesin, Ivan N. (2014). Birnbaum, Z. W.; Lukacs, E. (eds.).
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1074:. Vol. 1 (2nd ed.). Courier Dover Publications.
934:"Prizes Awarded by the Paris Academy of Sciences for 1914"
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followed up on this by formalizing what is now called the
569:
Approximation of the Riemann integral by rectangular areas
565:
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to measure functions whose domain of definition is not a
354:
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Henri LĂ©on Lebesgue (28 juin 1875 - 26 juillet 1941 )
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His lectures from 1902 to 1903 were collected into a "
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Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
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Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
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Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
597:. The first theory of integration was developed by
560:
625:, mathematicians felt that Newton's and Leibniz's
472:(x) increases to the limit f(x), the integral of f
979:
37:Not to be confused with the French palaeographer
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891:Obituary Notices of Fellows of the Royal Society
371:. In 1910 Lebesgue moved to the Sorbonne as a
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277:Henri Lebesgue was born on 28 June 1875 in
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1189:"L'intégration des fonctions non bornées"
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1098:Classical and Modern Integration Theories
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861:List of things named after Henri Lebesgue
601:in the 3rd century BC with his method of
383:. Henri Lebesgue died on 26 July 1941 in
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317:In 1894, Lebesgue was accepted at the
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888:(1944). "Henri Lebesgue. 1875-1941".
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801:Lebesgue's universal covering problem
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1361:Foreign members of the Royal Society
629:did not have a rigorous foundation.
1015:. Running Press. pp. 1041â87.
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1321:20th-century French mathematicians
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806:LebesgueâRokhlin probability space
549:. He also had a disagreement with
265:("Integral, length, area") at the
25:
1382:
1269:
944:(2358): 518â519. 7 January 1915.
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1262:from the original on 2009-09-16.
1219:from the original on 2014-08-05.
1176:from the original on 2009-09-16.
1045:. Infobase Publishing. pp.
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751:Lebesgue's decomposition theorem
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846:Dominated convergence theorem
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289:and his mother was a school
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652:and takes the limit of the
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285:. Lebesgue's father was a
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856:Tietze extension theorem
636:developed epsilon-delta
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373:maßtre de conférences
325:done at that time by
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1009:Hawking, Stephen W.
980:O'Connor, John J.;
517:Lipschitz condition
391:Mathematical career
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267:University of Nancy
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719:Henstock integral
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146:
142:
139:
135:
132:
128:
125:
121:
118:
115:
111:
107:
103:
94:July 26, 1941
93:
89:
85:
81:
77:
72:June 28, 1875
64:
60:
56:
51:
44:
40:
39:Henri LebĂšgue
33:
19:
1341:Intuitionism
1242:
1238:
1225:
1199:
1195:
1182:
1156:
1152:
1139:
1129:
1125:
1121:
1118:
1116:
1097:
1090:
1071:
1065:
1041:
1012:
1003:
989:
975:
960:
941:
937:
928:
895:
889:
851:Osgood curve
708:
685:
681:
658:
631:
607:Isaac Newton
581:
574:
540:
533:
529:
514:
486:
453:
447:
438:
430:
411:
405:
399:
362:
359:
316:
302:and then at
295:tuberculosis
276:
262:
240:
239:
183:Institutions
166:
96:(1941-07-26)
1311:1941 deaths
1306:1875 births
1291:(in French)
1245:: 255â257.
1159:: 191â250.
603:quadratures
583:Integration
551:Ămile Borel
408:Weierstrass
335:Ămile Borel
221:Paul Montel
211:Ămile Borel
177:Mathematics
113:Nationality
18:H. Lebesgue
1300:Categories
868:References
650:rectangles
599:Archimedes
589:under the
287:typesetter
68:1875-06-28
1202:: 71â92.
920:122854745
728:manifolds
724:real line
1257:Archived
1214:Archived
1171:Archived
1128:(1919),
1124:(1918),
1011:(2005).
734:See also
673:finitely
665:codomain
595:function
547:topology
508:and the
423:polygons
331:Sorbonne
279:Beauvais
245:ForMemRS
161:for 1914
76:Beauvais
967:at the
700:measure
688:measure
452:tract"
291:teacher
250:French:
1132:(1920)
1109:
1078:
1053:
1019:
938:Nature
918:
912:768841
910:
692:length
661:domain
640:, and
638:limits
464:, and
402:, 1904
173:Fields
151:Awards
117:French
106:France
84:France
1260:(PDF)
1235:(PDF)
1217:(PDF)
1192:(PDF)
1174:(PDF)
1149:(PDF)
916:S2CID
908:JSTOR
593:of a
591:graph
450:Borel
385:Paris
351:Nancy
312:Paris
102:Paris
1107:ISBN
1076:ISBN
1051:ISBN
1017:ISBN
654:sums
609:and
587:area
545:and
480:and
341:and
306:and
283:Oise
91:Died
80:Oise
62:Born
1247:doi
1204:doi
1161:doi
1047:164
946:doi
900:doi
355:PhD
310:in
1302::
1255:.
1243:37
1241:.
1237:.
1212:.
1200:36
1198:.
1194:.
1169:.
1157:35
1155:.
1151:.
1130:37
1126:36
1122:35
1115:.
1101:.
1049:.
1031:^
994:,
988:,
984:,
942:94
940:.
936:.
914:.
906:.
894:.
875:^
706:.
527:.
512:.
460:,
425:,
387:.
314:.
281:,
104:,
82:,
78:,
1249::
1206::
1163::
1084:.
1059:.
1025:.
954:.
948::
922:.
902::
896:4
579:.
474:n
470:n
415:.
248:(
70:)
66:(
34:.
20:)
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