Lebesgue's Theory of Integration: Its Origins and Development

Description

In this book, Hawkins elegantly places Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries. Hawkins was awarded the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for notable exposition and exceptional scholarship in the history of mathematics.

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This book is an introduction to the history of the Lebesgue integral, its main purpose being to appreciate what Lebesgue did seventy years ago, and why he did it. The first four chapters cover the hundred year period culminating with the appearance of the first edition of Lebesgue's famous book in 1904 [Leçons sur intégration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904; second edition, 1928]. What seems of greatest interest to teachers and students of real analysis is that explicit references are given to a list of classical errors and confusions, that would be inexcusable in a graduate student today: they were made by Ampère, Cauchy, Fourier, Galois, Gauss, Lagrange and by such lesser lights as du Bois-Reymond, Gilbert, Hankel, Königsberger, Lamarle, Lipschitz, all of whom were considered as mathematicians in their own right. In addition, the book refers to half a dozen "proofs'', in leading textbooks of the period, of the false proposition that every continuous function is differentiable. The relatively familiar and basic contributions of Cauchy, Riemann, Weierstrass and Cantor are recalled, but these can be read in more detail elsewhere. More emphasis is therefore naturally placed on the background researches of Darboux, Dini, Dirichlet, Harnack, Jordan and H. J. S. Smith, and on the whole atmosphere of a subject slowly progressing towards rigour. The remainder of the book contents itself with summarizing the ideas of Lebesgue's theory and some of the most immediate allied developments. This is historically less interesting since it says nothing of the polemic between Lebesgue and Borel, and does less than justice to the other mathematicians responsible for the ferment of ideas in the early 1900's. It also makes no mention, in its 300 or so references, of the way in which set theory and measure have subsequently gradually shed their naive and intuitive background; and it covers almost nothing of the material in the authoritative books of Saks and Carathéodory, in fact, almost nothing later than 1910, so that even Lebesgue's much enlarged second edition is not cited. A short paragraph, drawing attention to these matters and perhaps to the stochastic integrals of Wiener and others, or to the work of G. P. Tolstov, which brings the Denjoy and Lebesgue integrals so much closer to classical ideas, might have been helpful.

Reviewed by L. C. Young