By Thomas Hawkins

During this e-book, Hawkins elegantly areas Lebesgue's early paintings on integration idea inside in right historic context via concerning it to the advancements throughout the 19th century that influenced it and gave it value and in addition to the contributions made during this box via Lebesgue's contemporaries. Hawkins used to be provided the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for outstanding exposition and extraordinary scholarship within the heritage of arithmetic.

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**Additional resources for Lebesgue's Theory of Integration: Its Origins and Development **

**Sample text**

1 (1872-73), 121; 2 (1873-74), 66. 28 LEBESGUE'S THEORY OF INTEGRATION But term-by-term integration of the series over [0,x] yields n=1 f =un(t)dt 0 [e-ns:s _ e-(n+l)s==] = e-==. nal The series fails to converge uniformly because, as Darboux realized, co I Rn(1/n) I =2n/e, where Rn(x) = E uk(x). In other words, the series ksn is not uniformly bounded on [0,x]. ) But Darboux did not introduce the notion of uniform boundedness and apparently had no idea of its importance for term-byterm integration.

Although composed ten years after Riemann's researches, Lipschitz' paper was not influenced by them, and it appears doubtful that they were known to him. ) Lipschitz began by listing the possible ways in which a function could fail to satisfy the conditions under which Dirichlet had proved his theorem. One way was if a bounded and piecewise monotonic function f was discontinuous at an infinite number of points in the interval Lipschitz realized that Dirichlet's proof would apply if the integral concept could be extended to f, and he concluded that this could easily be accomplished for functions satisfying Dirichlet's condition on the set D of discontinuities (D is nowhere dense).

Continuity and integrability) usually ascribed to the more familiar examples of functions occurring in analysis, even though, as in his examples, the function may be represented by an analytical expression. One of his disciples, Hermann Hankel (1839-73), a professor of mathematics at the University of Tubingen, was particularly impressed by Riemann's viewpoint and discoursed upon it at length in two works composed at about the same time-an article "Grenze" in the Allgemeine Encyklopadie der Wissenschaften and Kunste [1871: 189-2061 and a lengthy essay, "Untersuchungen fiber die unendlich oft oszillierenden and unstetigen Funktionen" [1870].