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Infinite Dimensional Analysis A Hitchhiker’s Guide 3rd Edition

Charalambos D. Aliprantis Kim C. Border Infinite Dimensional Analysis A Hitchhiker’s Guide Third Edition With 38 Figures and 1 Table 123

Professor Charalambos D. Aliprantis Department of Economics Krannert School of Management Rawls Hall, Room 4003 Purdue University 100 S. Grant Street West Lafayette IN 47907-2076 USA E-mail: aliprantis@mgmt.purdue.edu Professor Kim C. Border California Institute of Technology Division of the Humanities and Social Sciences 228–77 1200 E. California Boulevard Pasadena CA 91125 USA E-mail: kcborder@caltech.edu Cataloging-in-Publication Data Library of Congress Control Number: 2006921177 ISBN-10 3-540-29586-0 3rd ed. Springer Berlin Heidelberg New York ISBN-13 978-3-540-29586-0 3rd ed. Springer Berlin Heidelberg New York ISBN 3-540-65854-8 2nd ed. Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provi- sions of the German Copyright Law of September 9, 1965, in its current version, and per- mission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 1999, 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner Production: Helmut Petri Printing: Strauss Offsetdruck SPIN 11572817 Printed on acid-free paper – 42/3153 – 5 4 3 2 1 0

In memoriam Yuri Abramovich Jeﬀrey Banks Taesung Kim Richard McKelvey . . . colleagues, collaborators, friends.

Preface to the third edition This new edition of The Hitchhiker’s Guide has beneﬁtted from the comments of many individuals, which have resulted in the addition of some new material, and the reorganization of some of the rest. The most obvious change is the creation of a separate Chapter 7 on convex analysis. Parts of this chapter appeared in elsewhere in the second edition, but much of it is new to the third edition. In particular, there is an expanded discussion of support points of convex sets, and a new section on subgradients of convex functions. There is much more material on the special properties of convex sets and functions in ﬁnite dimensional spaces. There are improvements and additions in almost every chapter. There is more new material than might seem at ﬁrst glance, thanks to a change in font that re- duced the page count about ﬁve percent. We owe a huge debt to Valentina Galvani, Daniela Puzzello, and Francesco Rusticci, who were participants in a graduate seminar at Purdue University and whose suggestions led to many improvements, especially in chapters ﬁve through eight. We particularly thank Daniela Puzzello for catching uncountably many errors throughout the second edition, and simplifying the statements of several theorems and proofs. In another graduate seminar at Caltech, many improvements and corrections were suggested by Joel Grus, PJ Healy, Kevin Roust, Maggie Penn, and Bryan Rogers. We also thank Gabriele Camera, Chris Chambers, John Duggan, Federico Echenique, Monique Florenzano, Paolo Ghirardato, Dionysius Glycopantis, Aviad Heifetz, John Ledyard, Fabio Maccheroni, Massimo Marinacci, Efe Ok, Uzi Segal, Rabee Tourky, and Nicholas Yannelis for their corrections and questions, their encouragement, and their (not always heeded) advice. Finally, we acknowledge our intellectual debt to our mentor Wim Luxemburg, and the constant support of the late Yuri Abramovich. Roko Aliprantis KC Border November 2005

viii Preface Preface to the second edition In the nearly ﬁve years since the publication of what we refer to as The Hitch- hiker’s Guide, we have been the recipients of much advice and many complaints. That, combined with the economics of the publishing industry, convinced us that the world would be a better place if we published a second edition of our book, and made it available in paperback at a more modest price. The most obvious diﬀerence between the second and the original edition is the reorganization of material that resulted in three new chapters. Chapter 4 col- lects many of the purely set-theoretical results about measurable structures such as semirings and σ-algebras. The material in this chapter is quite independent from notions of measure and integration, and is easily accessible, so we thought it should come sooner. We also divided the chapter on correspondences into two separate chapters, one dealing with continuity, the other with measurability. The material on measurable correspondences is more detailed and, we hope, better written. We also put many of the representation theorems into their own Chap- ter 14. This arrangement has the side eﬀect of forcing the renumbering of almost every result in the text, thus rendering the original version obsolete. We feel bad about that, but like Humpty Dumpty, we doubt we could put it back the way it was. The second most noticeable change is the addition of approximately seventy pages of new material. In particular, there is now an extended treatment of analytic sets in Polish spaces, which is divided among Sections 3.14, 12.5, and 12.6. There is also new material on Borel functions between Polish spaces in Section 4.11, a discussion of Lusin’s Theorem 12.8, and a more general treatment of the Kol- mogorov Extension Theorem in Section 15.6. There are many other additions through out the text, including a handful of additional ﬁgures. The truly neu- rotic reader may have noticed that by an almost unimaginable stroke of luck every chapter begins on a recto page. We revised the exposition of numerous proofs, especially those we could no longer follow. We also took the opportunity to expunge dozens of minor errors and misprints, as well as a few moderate errors. We hope that in the process we did not introduce too many new ones. If there are any major errors, neither we nor our students could ﬁnd them, so they remain. We thank Victoria Mason at Caltech and Werner Müller, our editor at Spring- er–Verlag, for their support and assistance. In addition to all those we thanked in the original edition, we are grateful for conversations (or email) with Jeﬀrey Banks, Paolo Battigalli, Owen Burkinshaw, John Duggan, Mark Fey, Paolo Ghirardato, Serena Guarnaschelli, Alekos Kechris, Antony Kwasnica, Michel Le Breton, John Ledyard, Massimo Marinacci, Jim

Preface ix Moore, Frank Page, Ioannis Polyrakis, Nikolaos Sofronidis, Rabee Tourky, Nick Yannelis, . . . and especially Yuri Abramovich for his constant encouragement and advice. Roko Aliprantis KC Border May 1999 Preface to the ﬁrst edition This text was born out of an advanced mathematical economics seminar at Cal- tech in 1989–90. We realized that the typical graduate student in mathematical economics has to be familiar with a vast amount of material that spans several traditional ﬁelds in mathematics. Much of the material appears only in esoteric research monographs that are designed for specialists, not for the sort of gener- alist that our students need be. We hope that in a small way this text will make the material here accessible to a much broader audience. While our motivation is to present and organize the analytical foundations underlying modern economics and ﬁnance, this is a book of mathematics, not of economics. We mention appli- cations to economics but present very few of them. They are there to convince economists that the material has some relevance and to let mathematicians know that there are areas of application for these results. We feel that this text could be used for a course in analysis that would beneﬁt mathematicians, engineers, and scientists. Most of the material we present is available elsewhere, but is scattered throughout a variety of sources and occasionally buried in obscurity. Some of our results are original (or more likely, independent rediscoveries). We have included some material that we cannot honestly say is necessary to understand modern economic theory, but may yet prove useful in future research. On the other hand, we wished to ﬁnish this work in our children’s lifetimes, so we have not presented everything we know, or everything we think that you should learn. You should not conclude that we feel that omitted topics are unimportant. For instance, we make no mention of diﬀerentiability, although it is extremely important. We would like to promise a second volume that would address the shortcomings of this one, but the track record of authors making such promises is not impressive, so we shall not bother. Our choice of material is a bit eccentric and reﬂects the interaction of our tastes. With apologies to D. Adams [4] we have compiled what we like to describe as a hitchhiker’s guide, or low budget touring guide, to analysis. Some of the

x Preface areas of analysis we explore leisurely on foot (others might say in a pedestrian fashion), other areas we pass by quickly, and still other times we merely point out the road signs that point to interesting destinations we bypass. As with any good hitchhiking adventure, there are detours and probably wrong turns. We have tried to write this book so that it will be useful as both a reference and a textbook. We do not feel that these goals are antithetical. This means that we sometimes repeat ourselves for the beneﬁt of those who start in the middle, or even at the end. We have also tried to cross-reference our results as much as possible so that it is easy to ﬁnd the prerequisites. While there are no formal exercises, many of the proofs have gaps indicated by the appearance of the words “How” and “Why.” These should be viewed as exercises for you to carry out. We seize this opportunity to thank Mike Maxwell for his extremely consci- entious job of reading the early drafts of this manuscript. He caught many er- rors and obscurities, and substantially contributed to improving the readability of this text. Unfortunately, his untimely graduation cut short his contributions. We thank Victoria Mason for her valuable support and her catering to our ec- centricities. We give special thanks to Don Brown for his moral support, and to Richard Boylan for nagging us to ﬁnish. We also thank Wim Luxemburg for his enlightening conversations on diﬃcult issues, and for sharing his grasp of history. We acknowledge beneﬁcial conversations with Yuri Abramovich, Owen Burkin- shaw, Alexander Kechris, Taesung Kim, and Nick Yannelis. We thank the partic- ipants in the seminar at Caltech: Richard Boylan, Mahmoud El-Gamal, Richard McKelvey, and Jeﬀ Strnad. We also express our gratitude to the following for working through parts of the manuscript and pointing out errors and suggesting improvements: Kay-yut Chen, Yan Chen, John Duggan, Mark Fey, Julian Jami- son, John Ledyard, Katya Sherstyuk. Michel Le Breton and Lionel McKenzie prompted us to include some of the material that is here. We thank Werner Müller, our editor at Springer–Verlag, for his eﬃciency and support. We typed and typeset this text ourselves, so we truly are responsible for all errors—mathematical or not. Don’t Panic Roko Aliprantis KC Border May 1994

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