From: Bill Dubuque Subject: Re: I am losing my math ability... Date: 08 May 1999 03:19:27 -0400 Newsgroups: sci.math Hankel O'Fung wrote: | | I graduated from university almost ten years ago. I don't remember much | mathematics now. Now I wish to learn some integration theory (Lebesgue, | Riemann-Stieltjes, Lebesgue-Stieltjes, Perron), and want some books that | explain the rationale behind why some integrals are defined in the way | they are so defined. In other words, I wish to read a book which is | inspiring, but not containing only theorems and proofs. Any suggestions? You might find the following of interest: ------------------------------------------------------------------------------ 885.01002 Pier, Jean-Paul Histoire de l'integration. Vingt-cinq siecles de mathematiques. (French) (History of integration. 25 centuries of mathematics). [B] Culture Scientifique. Paris: Masson. x, 306 p. (1996). [ISBN 2-225-85324-X/pbk] To cover integration from Hippocrates and Archimedes to Henstock, Kurzweil, nonstandard methods, and Feynman on a few hundred pages must appear a next to impossible task. This remarkable book is successful in several respects. It may be used as an encyclopedia of the history of integration, of its main figures and its major developments. It shows why and how new ideas and conceptions entered the scene. With profound understanding of history it illuminates a significant subject from a present day point of view. Much space is given to quotations, in the original languages if Latin, English, French, German or Italian, both from original sources and from reliable secondary texts. All of the 1200 titles of the carefully composed bibliography are referred to in the main text. Yet, the book, and every single chapter, are delightful and exciting reading, even when taken as fairly accessible introductions to the leading ideas of, say, Lebesgue, the `overall invasion of probability theory', or the Henstock-Kurzweil (generalized Riemann) integral. The author does not hesitate to include well-founded evaluations, either of his own or through quotations, perhaps too many of them taken from Bourbaki and Dieudonne. Selections had to be made. One may regret that pre-Newtonian integration is confined to 20 pages, little more are given to harmonic analysis of the twentieth century. Indeed there is little want for dwelling on trodden ground, and more demand for a comprehensive presentation of the post-Lebesgue era. Still, it is instructive to compare the author's presentation of earlier centuries with that given in Encyklopaedie Math. Wiss. around 1900 by A. Voss et al., unfortunately not included in the bibliography here. Another text that is missing here is {\it J. Luetzen}, The prehistory of the theory of distributions. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 494.46038). Near the end of the present century some need may be felt for a unifying recast of the diverging theories of integration. This feeling is present here only in a subliminal way. More explicitly it is expressed by {\it R. G. Bartle} [Am. Math. Mon. 103, No. 8, 625-632 (1996; Zbl 884.26007)]. There is an index of names (but no subject index) and an historical index (names with dates of life). -- The book will become a must for historians of mathematics, and can be highly recommended to students and to mathematicians working in analysis. An English translation seems worthwhile. [ D.Laugwitz (Darmstadt) ] ------------------------------------------------------------------------------ 98k:01003 01A05 (26-03 28-03 43-03 60-03) Pier, Jean-Paul (LUX-CUL) Histoire de l'intégration. (French) [History of integration] Vingt-cinq siècles de mathématiques. [Twenty-five centuries of mathematics] Culture Scientifique. [Scientific Education] Masson, Paris, 1996. xii+307 pp. 245 F. ISBN 2-225-85324-X ------------------------------------------------------------------------------ This book contains a comprehensive historical account of the concept of the integral and related ideas. The first chapter describes contributions from ancient Greece, China and the Arabs, and the second considers European contributions up to the mid-17th century. The next two chapters are concerned, respectively, with Newton and Leibniz. After an outline of 18th-century developments comes a chapter on Cauchy, covering not only his definition of the integral but also his work on existence theorems. This is followed by a chapter on 19th-century work, including the Riemann and Stieltjes integrals, and the first approaches to a theory of measure for point sets in Euclidean space (Cantor, Jordan, Peano). A special chapter is devoted to the work of Lebesgue and his immediate successors (Vitali, Fubini, de La Vallee Poussin). There follows an account of various developments up to about 1960; among the topics dealt with here are the Riesz-Fischer theorem and its consequences, the Denjoy and Perron integrals, measure and integration on abstract sets, vector-valued integrals, the Bourbaki approach to the integral, and the work of de Rham, Whitney and others in the context of differential geometry. A separate chapter is devoted to harmonic analysis and related topics (convolutions, invariant measures, distributions, etc.), and another to probability theory and its interpretation as a branch of measure theory, with some mention of ergodic theory and stochastic processes. A final chapter is concerned with developments over the period from 1960 to about 1994; among the topics referred to here are Hausdorff measure, the Henstock-Kurzweil integral, fuzzy sets, nonstandard analysis and noncommutative geometry. There is a massive bibliography, containing more than 800 items. A special feature of the book consists in the numerous quotations scattered through the text; sometimes these are drawn from the author under discussion, sometimes from later writers in the same field and sometimes from other historians. The book will be a valuable guide through the vast literature of the subject. So many detailed references are given that the text cannot be described as easy reading, though the task is lightened to some extent by the numerous quotations. \{Reviewer's remarks: The final displayed formula on p. 20 is apt to confuse the reader. There is some confusion between George D. Birkhoff and Garrett Birkhoff on p. 178 and in the bibliography (items B36 and B37). In the items relating to Cauchy on pp. 264--5, the series number ((1) or (2)) has been omitted in the references to his Oeuvres completes.\} Reviewed by F. Smithies ------------------------------------------------------------------------------ 884.26007 Bartle, Robert G. Return to the Riemann integral. [J] Am. Math. Mon. 103, No.8, 625-632 (1996). [ISSN 0002-9890] The author made a case for introducing the generalized Riemann integral at the undergraduate level. The integral is also known as the gauge integral or the Kurzweil-Henstock integral, which includes properly the Riemann, Lebesgue, and the improper Riemann integral. The monotone convergence theorem and the dominated convergence theorem are valid for the generalized Riemann integral. Also it has a simple version of the fundamental theorem. The link with measure theory is also possible. Finally, the author surveyed the books available on this integral. [ Lee Peng-Yee (Singapore) ] ------------------------------------------------------------------------------ 94d:01039 01A60 (01A55 22-03 26-03 28-03) Michel, Alain (F-PROV) Constitution de la théorie moderne de l'intégration. (French) [Formation of the modern theory of integration] Mathesis. Librairie Philosophique J. Vrin, Paris, 1992. 338 pp. 212 F. ISBN 2-7116-1064-0 ------------------------------------------------------------------------------ Integration theory has one of the oldest histories in mathematics, two essential highlights being represented by the invention of calculus and Lebesgue's revolution at the beginning of the XXth century. The book under review covers the background for the "modern" theory from Cauchy to Lebesgue and Daniell. It provides a remarkably argued study of the development of concepts leading to the interpretation of the integral in terms of a linear functional by Riesz, Stone, and Bourbaki; all kinds of aspects of the evolution of ideas up to the "modern" theory are examined. The historical treatment constitutes a median way between looking for explanations through queries raised by practical problems or motivations, and tracing "philosophical" origins demanding a proper intrinsic analysis. We briefly give the feeling of the contents of the 13 dense chapters of the book. Cauchy pinpoints the definite integral; as a linear functional it acquires a logically primary status (I). Riemann extends considerations to discontinuous functions (II). Connections with measures arise in the works of Jordan and Borel (III). Lebesgue dissociates the notions of primitive and indefinite integral. Whereas Riemann's procedure is still ruled by the intuition of continuity, Lebesgue makes integration of discontinuous functions "natural"; the convergence theorem assures the superiority of his methods (IV). Absolute continuity, actually introduced by Vitali, and complete additivity become important features in Lebesgue's theory. Young introduces a procedure essentially equivalent to Lebesgue's (V). The Stieltjes integral provides rigor for the physical notion of moment; the Lebesgue-Stieltjes integral is the backbone of the whole integration system. The Riesz representation theorem inaugurates the contemporary era of functional analysis (VI). Radon launches abstract integration theories; Frechet comes to a full understanding of the method. Nikodym complements Radon's results. Daniell succeeds in giving an elegant presentation of Lebesgue's theory, bypassing considerations of measures. Bourbaki chooses Radon measure as the fundamental concept (VII). The consideration of invariance properties for integration by Riemann and Poincare leads to the birth of analysis situs (VIII). Weyl introduces integration on Lie groups (IX). Haar constructs an invariant integral on a locally compact separable group (X). Existence and uniqueness of an invariant integral on a general locally compact group are established by Weil. For Michel, the topology of integration theory is therefore mainly the topology of compactness (XI). Harmonic analysis then grows out of the work of Pontryagin (XII). Hilbert's spectral theorem, Stone's formalism and von Neumann's spectral theorem implement the integral spectral theory (XIII). Of course, the richly documented, comprehensive presentation of this treatise could not be exhaustive. One may have asked for more details concerning measure-theoretical approaches, such as those due to Caratheodory or Halmos, connections with probability and ergodicity, Denjoy's and Perron's methods, Schwartz's distribution theory, and Connes' noncommutative integration. Feynman's integral, the works of Kurzweil and Henstock, and nonstandard integration are not mentioned. This outstanding reference source benefits from an original presentation. It is indeed a most valuable tool for the understanding of the evolution of one of the most important mathematical concepts up to the 1960s at least. Reviewed by Jean-Paul Pier ------------------------------------------------------------------------------ 577.01001 Nikiforovskij, V.A. The way to the integral. (Put' k integralu). (Russian) [B] Seriya: ''Istoriya Nauki i Tekhniki''. Moskva: ''Nauka''. 192 p. R. 0.65 (1985). This useful booklet gives a survey of the development of the integration theory up to the end of the nineteenth century. It consists of six chapters. The first chapter deals very shortly with ''Sources'' that is to say with Greek mathematicians living before Archimedes as Eudoxus and Euclid. The second chapter, called ''Birth of an idea'', explains the integration theory of Archimedes. The third chapter ''Development of an idea'' is mainly concerned with Kepler, Cavalieri, the Italian (Galilei, Torricelli) and French school (Fermat, Roberval, Pascal), and the English mathematicians Wallis and Barrow. The fourth chapter is called ''Origin and constitution of the integral''. Newton's and Leibniz's achievements, the contributions of the Bernoullis and Euler are discussed. The fifth chapter is dedicated to Cauchy and Riemann. The last chapter gives some hints to the further development, especially to Lebesgue and the Russian mathematicians Kolmogorov, Lusin and others. [ E.Knobloch ] ------------------------------------------------------------------------------ 584.26005 Henstock, Ralph The Lebesgue syndrome. [J] Real Anal. Exch. 9, 96-110 (1984). [ISSN 0147-1937] This talk is partly historical, partly personal, and partly a looking to the future. The cornerstone in the development on the way of evolution of integrals theories is the Lebesgue's work which is the culmination of all that had gone before. Although this was a giant step forward, soon became clear that it was not good enough to integrate all derivatives, and Denjoy gave his construction - what is now known as the Denjoy-Perron integral. Lebesgue showed that his integral is the limit of Riemann sums, and so did Denjoy, though neither gave anything explicit. The first explicit construction for non-negative functions was given by Beppo Levi. The personal side of this talk explains the author's contribution at this stage: ''I was turned upside down in 1958, throwing away Lebesgue and gresping Riemann''. Returning to Riemann sums but with a more general limit he gets the so called generalized Riemann integral or Riemann-complete integral. This work was independent of J. Kurzweil's paper in 1957. The new integral is more practical and easier than most Lebesgue theories, and, before all, it is a non-absolute theory (the difference being analogous to the difference between all convergent series and the absolutely convergent ones). The Lebesgue syndrome is the ignorance with respect to the work done on non-absolute integration. How large is this syndrome? It may be measured by comparing the number of papers reviewed in one year in M.R.: about 200 on absolute theory against 6 devoted to non-absolute integration. As a consequence, the author throw out a challenge to look at the papers using Lebesgue theory to see whether the proofs can be improved and the contents generalized by generalized Riemann methods. [ J.Benkoe ] ------------------------------------------------------------------------------ 84m:01027 01A55 (01A60 26-03 28-03) Knobloch, Eberhard (D-TUB) Von Riemann zu Lebesgue---zur Entwicklung der Integrationstheorie. (German. English, French summary) [From Riemann to Lebesgue--- on the development of integration theory] Historia Math. 10 (1983), no. 3, 318--343. ------------------------------------------------------------------------------ The author traces not only the road from Riemann to Lebesgue that is indicated in his title but also the prehistory from Cauchy and Dirichlet. This is one part of the history of mathematics which has been quite well examined; the author adds no essentially new perspectives or insights, although his account is distinguished by a combination of clear outline and pertinent detailing. An excellent bibliography of primary and secondary literature is appended. Reviewed by I. Grattan-Guinness ------------------------------------------------------------------------------ 58 #28384 28-03 (01A60) {\cyr Medvedev, F. A.} {\cyr Razvitie ponyatiya integrala.} (Russian) [Development of the concept of integral] Izdat. ``Nauka'', Moscow, 1974. 423 pp. 2.01 r. ------------------------------------------------------------------------------ Publisher's description: "This monograph is a presentation of the development of the notion of integral from the early beginnings of integration methods to the Lebesgue-Stieltjes integral. The presentation is closely related to the development of mathematical analysis and its applications; various generalizations of the notion of integral are seen as necessary consequences of the development of analysis and the theory of functions." Reviewed by Publisher's description ------------------------------------------------------------------------------ 42 #4368 01.40 (28.00) Hawkins, Thomas Lebesgue's theory of integration. Its origins and development. The University of Wisconsin Press, Madison, Wis.-London 1970 xv+227 pp. ------------------------------------------------------------------------------ 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 [Lecons sur integration 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 Ampere, Cauchy, Fourier, Galois, Gauss, Lagrange and by such lesser lights as du Bois-Reymond, Gilbert, Hankel, Konigsberger, 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 Caratheodory, 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 ------------------------------------------------------------------------------ 613.28001 Kupka, Joseph. Measure theory: the heart of the matter. [J] Math. Intell. 8, No.4, 47-56 (1986). [ISSN 0343-6993] This is an excellent exposition of the evolution of geometric measure theory from Euclid to Lebesgue. The ''heart of the matter'' referred to in the title is what the author calls the ''paving stone technique'' - the idea of measuring complicated or irregular areas by exhausting or covering them with simple areas whose measurements are known. The author describes and explains very clearly the different ways in which the paving stone technique has been used in Euclid's ''Elements of Geometry'' and later on to extend classes of measurable areas and of integrable functions until geometric measure theory reached its maturity with Lebesgue's construction of measure and integral. The author also emphasizes the role of the paving stone idea in probability theory, and he briefly discusses the importance of abstract measure theory developed after Lebesgue in the foundations of probability theory and mathematical statistics. [ Klaus D.Schmidt ]