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Bibliographical and Historical Comments One gets a strange feeling having seen the same drawings as if drawn by the same hand in the works of four schol- ars that worked completely independently of each other. An involuntary thought comes that such a striking, myste- rious activity of mankind, lasting several thousand years, cannot be occasional and must have a certain goal. Having acknowledged this, we come by necessity to the question: what is this goal? I.R. Shafarevich. On some tendencies of the develop- ment of mathematics. However, also in my contacts with the American Shake- speare scholars I confined myself to the concrete problems of my research: dating, identification of prototypes, direc- tions of certain allusions. I avoided touching the problem of personality of the Great Bard, the “Shakespeare prob- lem”; neither did I hear those scholars discussing such a problem between themselves. I.M. Gililov. A play about William Shakespeare or the Mystery of the Great Phoenix. The extensive bibliography in this book covers, however, only a small portion of the existing immense literature on measure theory; in particular, many authors are represented by a minimal number of their most character- istic works. Guided by the proposed brief comments and this incomplete list, the reader, with help of modern electronic data-bases, can considerably en- large the bibliography. The list of books is more complete (although it cannot pretend to be absolutely complete). For the reader’s convenience, the bibli- ography includes the collected (or selected) works of A.D. Alexandrov [15], R. Baire [47], S. Banach [56], E. Borel [114], C. Carath´eodory [166], A. Den- joy [215], M. Fr´echet [321], G. Fubini [333], H. Hahn [401], F. Haus- dorff [415], S. Kakutani [482], A.N. Kolmogorov [535], Ch.-J. de la Vall´ee Poussin [575], H. Lebesgue [594], N.N. Lusin [637], E. Marczewski [652], J. von Neumann [711], J. Radon [780], F. Riesz [808], V.A. Rohlin [817], W. Sierpi´nski [881], L. Tonelli [956], G. Vitali [990], N. Wiener [1017], and G. &W. Young [1027], where one can find most of their cited works along with other papers related to measure theory. Many works in the bibliography 410 Bibliographical and Historical Comments are only cited in the main text in connection with concrete results (including exercises and hints). Some principal results are accompanied by detailed com- ments; in many other cases we mention only the final works, which should be consulted concerning the previous publications or the history of the question. Dozens of partial results mentioned in the book have an extremely interesting history, revealed through the reading of old journals, the exposition of which I had to omit with regret. Most of the works in the bibliography are in English and French; a rel- atively small part of them (in particular, some old classical works) are in German, Russian, and Italian. For most of the Russian works (excepting a limited number of works from the 1930s–60s), translations are indicated. The reader is warned that in such cases, the titles and author names are given according to the translation even when versions more adequate and closer to the original are possible. Apart from the list of references, I tried to be consistent in the spelling of such names as Prohorov, Rohlin, Skorohod, and Tychonoff, which admit different versions. The letter “h” in such names is responsible for the same sound as in “Hardy” or “Halmos”, but in different epochs was transcribed differently, depending on to which foreign language (French, German, or English) the translation was made. Nowadays in official documents it is customary to represent this “h” in the Russian family names as “kh” (although, it seems, just “h” would be enough). Now several remarks are in order on books on Lebesgue measure and in- tegration. The first systematic account of the theory was given by Lebesgue himself in the first edition of his lectures [582] in 1904. In 1907, the first edition of the fundamental textbook by Hobson [436] was published, where certain elements of Lebesgue’s theory were included (in later editions the cor- responding material was considerably reworked and enlarged); next the books by de la Vall´ee Poussin [572] (note that in later editions the Lebesgue integral is not considered) and [574] and Carath´eodory [164] appeared. It is worth noting that customarily the form La Vall´ee Poussin de is used for the alpha- betic ordering; however, in some libraries this author is to be found under “V” or “P”, see Burkill [149]. These four books are frequently cited in many works of the first half of the 20th century. Let us also mention an extensive treatise Pierpont [756]. Some elements of Lebesgue’s measure theory were discussed in Hausdorff [412] (in later editions this material was excluded). Some back- ground was given in Sch¨onflies[858]. Elements of Lebesgue’s measure theory were considered in the book Nekrasov [709] published in 1907. Early surveys of Lebesgue’s theory were La Vall´eePoussin [573], Bliss [95], Hildebrandt [432], and a series of articles Borel, Zoretti, Montel, Fr´echet [115], published in the Encyclopedie des sciences math´ematiques (the reworked German ver- sion was edited by Rosenthal [823]). It is worth mentioning that in Lusin’s classical monograph [633], the first edition of which was published in 1915 and was his magister dissertation (by a special decision of the scientific committee, the degree of Doctor was conferred on Lusin in recognition of the outstanding level of his dissertation), the fundamentals of Lebesgue’s theory were assumed Bibliographical and Historical Comments 411 to be known (references were given to the books by Lebesgue and de la Vall´ee Poussin). The subject of Lusin’s dissertation was the study of fine properties of the integral (not only the Lebesgue one, but also more general ones), the primitives and trigonometric series. Another very interesting document is the magister dissertation of G.M. Fichtenholz [288] (the author of the excellent calculus course [295]) completed in February 1918. Unfortunately, due to the well-known circumstances of the time, this remarkable handwritten man- uscript was never published and was not available to the broad readership.1 Fichtenholz’s dissertation is a true masterpiece, and many of its results, still not widely known, retain an obvious interest. The manuscript contains 326 pages (the title page is posted on the website of the St.-Petersburg Mathe- matical Society; the library of the Department of Mechanics and Mathematics of Moscow State University has a copy of the dissertation). The introduction (pp. 1–58) gives a concise course on Lebesgue’s integration. The principal original results of G.M. Fichtenholz are concerned with limit theorems for the integral and are commented on in appropriate places below (see also Bogachev [106]). The dissertation contains an extensive bibliography (177 titles) and a lot of comments (in addition to historical notes, there are many interesting remarks on mistakes or gaps in many classical works). In the 1920s the following books appeared: Hahn [398], Kamke [485], van Os [731], Schlesinger, Plessner [853], Townsend [963]. Vitali’s books [988], [989] also contain large material on Lebesgue’s integration. In 1933, the first French edition of the classical book Saks [840] was published (the second edition was published in English in 1937); this book still remains one of the most influential reference texts in the subject. The same year was marked by publication of Kolmogorov’s celebrated monograph [532], which built math- ematical probability theory on the basis of abstract measure theory. This short book (of a booklet size), belonging to the most cited scientific works of the 20th century, strongly influenced modern measure theory and became one of the reasons for its growing popularity. Also in the 1930s, the textbooks by Titchmarsh [947], Haupt, Aumann [411] (the first edition), and Kestel- man [504] were published. Fundamentals of Lebesgue measure and integra- tion were given in Alexandroff, Kolmogorov [17]. The basic results of measure theory were presented in the book Tornier [961] on foundations of probability theory, which very closely followed Kolmogorov’s approach (a drawback of Tornier’s book is a complete omission of indications to the authorship of the presented theorems). In addition, in those years there existed lecture notes published later (e.g., von Neumann [710], Vitali, Sansone [991]). Note also the book Stone [914] containing material on the theory of integration. In 1941 the excellent book Natanson [706] was published (I.P. Natanson was Fichtenholz’s student and his book was obviously influenced by the aforemen- tioned dissertation of Fichtenholz). In McShane [668], the presentation of the 1I am most grateful to V.P. Havin, the keeper of the manuscript, for permission to make a copy, and to M.I. Gordin and A.A. Lodkin for their generous help. 412 Bibliographical and Historical Comments theory of the integral is based on the Daniell approach, and then a standard course is given including a chapter on the Lebesgue–Stieltjes integral. Jessen’s book [465] was composed of a series of journal expositions published in the period 1934–1947. Let us also mention Cramer’s book [190] on mathematical statistics where a solid exposition of measure and integration was included. It should be noted that Kolmogorov’s concept of foundations of probability theory lead to a deep penetration of the apparatus of general measure theory also into mathematical statistics, which is witnessed not only by Cramer’s book, but also by many subsequent expositions of the theoretical foundations of mathematical statistics, see Barra [62], Lehmann [600], Schmetterer [854]. After World War II the number of books on measure theory consider- ably increased because this subject became part of the university curriculum.
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  • Burgin M. Hypernumbers and Extrafunctions.. Extending The

    Burgin M. Hypernumbers and Extrafunctions.. Extending The

    SpringerBriefs in Mathematics SpringerBriefs in Mathematics Series Editors Krishnaswami Alladi Nicola Bellomo Michele Benzi Tatsien Li Matthias Neufang Otmar Scherzer Dierk Schleicher Benjamin Steinberg Yuri Tschinkel Loring W. Tu George Yin Ping Zhang SpringerBriefs in Mathematics showcase expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. For further volumes: http://www.springer.com/series/10030 Mark Burgin Hypernumbers and Extrafunctions Extending the Classical Calculus Mark Burgin Department of Mathematics University of California Los Angeles, CA 90095, USA ISSN 2191-8198 ISSN 2191-8201 (electronic) ISBN 978-1-4419-9874-3 ISBN 978-1-4419-9875-0 (eBook) DOI 10.1007/978-1-4419-9875-0 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012935428 Mathematics Subject Classification (2010): 46A45, 46G05, 46G12, 46H30, 46F10, 46F30 # Mark Burgin 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.