http://dx.doi.org/10.1090/gsm/045
An Introductio n to Measur e an d Integratio n This page intentionally left blank An Introductio n to Measur e an d Integratio n
SECOND EDITION
Inder K. Rana
Graduate Studies in Mathematics
Volume 45
;S!HSSJ»0 American Mathematical Society Providence, Rhode Island Editorial Board Steven G. Krantz David Saltman (Chair) David Sattinger Ronald Stern
2000 Mathematics Subject Classification. Primary 28-01; Secondary 28A05, 28A10, 28A12, 28A15, 28A20, 28A25, 28A33, 28A35, 26A30, 26A42, 26A45, 26A46.
ABSTRACT. This text presents a motivated introduction to the theory of measure and integration. Starting with an historical introduction to the notion of integral and a preview of the Riemann integral, the reader is motivated for the need to study the Lebesgue measure and Lebesgue integral. The abstract integration theory is developed via measure. Other basic topics discussed in the text are Pubini's Theorem, Lp-spaces, Radon-Nikodym Theorem, change of variables formulas, signed and complex measures.
Library of Congress Cataloging-in-Publication Data Rana, Inder K. An introduction to measure and integration / Inder K. Rana.—2nd ed. p. cm. — (Graduate texts in mathematics, ISSN 1065-7339 ; v. 45) Includes bibliographical references and index. ISBN 0-8218-2974-2 (alk. paper) 1. Lebesgue integral. 2. Measure theory. I. Title. II. Graduate texts in mathematics ; 45.
QA312 .R28 2002 515/.42—dc21 2002018244
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publica• tion is permitted only under license from Narosa Publishing House. Requests for such permis• sion should be addressed to Narosa Publishing House, 6 Community Centre, Panchscheel Park, New Delhi 110 017, India. First Edition © 1997 by Narosa Publishing House. Second Edition © 2002 by Narosa Publishing House. All rights reserved. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 In memory of my father Shri Omparkash Rana (24th April, 1924-26th May, 2002) This page intentionally left blank Contents
Preface xi
Preface to the Second Edition xvii
Recipe for a one semester course and interdependence of the chapters xix
Notations used in the text xxi
Prologue. The length function 1
Chapter 1. Riemann integration 5 §1.1. The Riemann integral: A review 5 §1.2, Characterization of Riemann integrable functions 18 §1.3. Historical notes: The integral from antiquity to Riemann 30 §1.4. Drawbacks of the Riemann integral 36
Chapter 2. Recipes for extending the Riemann integral 45 §2.1. A function theoretic view of the Riemann integral 45 §2.2. Lebesgue's recipe 47 §2.3. Riesz-Daniel recipe 49
Chapter 3. General extension theory 51 §3.1. First extension 51 §3.2. Semi-algebra and algebra of sets 54 vm Contents
§3.3. Extension from semi-algebra to the generated algebra 58 §3.4. Impossibility of extending the length function to all subsets of the real line 61 §3.5. Countably additive set functions on intervals 62 §3.6. Countably additive set functions on algebras 64 §3.7. The induced outer measure 70 §3.8. Choosing nice sets: Measurable sets 74 §3.9. The cr-algebras and extension from the algebra to the generated cr-algebra 80 §3.10. Uniqueness of the extension 84 §3.11. Completion of a measure space 89 Chapter 4. The Lebesgue measure on R and its properties 95 §4.1. The Lebesgue measure 95 §4.2. Relation of Lebesgue measurable sets with topologically nice subsets of R 99 §4.3. Properties of the Lebesgue measure with respect to the group structure on R 103 §4.4. Uniqueness of the Lebesgue measure 106 §4.5. * Cardinalities of the a-algebras C and B^ HO §4.6. Nonmeasurable subsets of R 113 §4.7. The Lebesgue-Stieltjes measure 114 Chapter 5. Integration 117 §5.1. Integral of nonnegative simple measurable functions 118 §5.2. Integral of nonnegative measurable functions 122 §5.3. Intrinsic characterization of nonnegative measurable functions 130 §5.4. Integrable functions 143 §5.5. The Lebesgue integral and its relation with the Riemann integral 153 §5.6. I/i[a, b] as completion of 1Z[a, b] 158 §5.7. Another dense subspace of L\[a, b] 163 §5.8. Improper Riemann integral and its relation with the Lebesgue integral 168 Contents IX
§5.9. Calculation of some improper Riemann integrals 172
Chapter 6. Fundamental theorem of calculus for the Lebesgue integral 175 §6.1. Absolutely continuous functions 175 §6.2. Differentiability of monotone functions 179 §6.3. Fundamental theorem of calculus and its applications 191
Chapter 7. Measure and integration on product spaces 209 §7.1. Introduction 209 §7.2. Product of measure spaces 212 §7.3. Integration on product spaces: Fubini's theorems 221 §7.4. Lebesgue measure on M2 and its properties 229 §7.5. Product of finitely many measure spaces 237
Chapter 8. Modes of convergence and Lp-spaces 243 §8.1. Integration of complex-valued functions 243 §8.2. Convergence: Point wise, almost everywhere, uniform and almost uniform 248 §8.3. Convergence in measure 255 §8.4. Lp-spaces 261
§8.5. *Necessary and sufficient conditions for convergence in Lp 270
§8.6. Dense subspaces of Lp 279 §8.7. Convolution and regularization of functions 281
§8.8. LOO(X,S,IJ,): The space of essentially bounded functions 291 §8.9. 1/2(X, 5,/i): The space of square integrable functions 296 §8.10. Z/2-convergence of Fourier series 306 Chapter 9. The Radon-Nikodym theorem and its applications 311 §9.1. Absolutely continuous measures and the Radon-Nikodym theorem 311 §9.2. Computation of the Radon-Nikodym derivative 322 §9.3. Change of variable formulas 331 Chapter 10. Signed measures and complex measures 345 X Contents
§10.1. Signed measures 345 §10.2. Radon-Nikodym theorem for signed measures 353 §10.3. Complex measures 365 r §10.4. Bounded linear functionals on L p(X, 5, /i) 373
Appendix A. Extended real numbers 385
Appendix B. Axiom of choice 389
Appendix C. Continuum hypothesis 391
Appendix D. Urysohn's lemma 393
Appendix E. Singular value decomposition of a matrix 395
Appendix F. Functions of bounded variation 397
Appendix G. Differentiable transformations 401
References 409
Index 413
Index of notations 419 Preface
"Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is un• suitable for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isola• tion and atrophy of mathematics. It seems extremely important that students and instructors should be protected from smug purism."
Richard Courant and Fritz John (Introduction to Calculus and Analysis)
This text presents a motivated introduction to the subject which goes under various headings such as Real Analysis, Lebesgue Measure and Integration, Measure Theory, Modern Analysis, Advanced Analysis, and so on. The subject originated with the doctoral dissertation of the French mathematician Henri Lebesgue and was published in 1902 under the ti• tle Integrable, Longueur, Aire. The books of C. Caratheodory [8] and [9], S. Saks [35], LP. Natanson [27] and P.R. Halmos [14] presented these ideas in a unified way to make them accessible to mathematicians. Because of its fundamental importance and its applications in diverse branches of mathe• matics, the subject has become a part of the graduate level curriculum. Historically, the theory of Lebesgue integration evolved in an effort to remove some of the drawbacks of the Riemann integral (see Chapter 1). However, most of the time in a course on Lebesgue measure and integra• tion, the connection between the two notions of integrals comes up only
XI Xll Preface after about half the course is over (assuming that the course is of one se• mester). In this text, after a review of the Riemann integral, the reader is acquainted with the need to extend it. Possible methods to carry out this extension are sketched before the actual theory is presented. This approach has given satisfying results to the author in teaching this subject over the years and hence the urge to write this text. The nucleus for the text was provided by the lecture notes of the courses I taught at Kurukshetra Uni• versity (India), University of Khartoum (Sudan), South Gujarat University (India) and the Indian Institute of Technology Bombay (India). These notes were slowly augmented with additional material so as to cover topics which have applications in other branches of mathematics. The end product is a text which includes many informal comments and is written in a lecture-note style. Any new concept is introduced only when it is needed in the logical development of the subject and it is discussed informally before the exact definition appears. The subject matter is developed by motivating examples and probing questions, as is normally done while teaching. I have tried to avoid slick proofs. Often a proof is either divided into steps or is presented in such a way that the main ideas of the proof emerge before the details follow.
Summary of the text The text opens with a Prologue on the length function and its properties which are basic for the development of the subject. Chapter 1 begins with a detailed review of the Riemann integral and its properties. This includes Lebesgue's characterization of Riemann integrable functions. It is followed by a brief discussion on the historical development of the integral from antiquity (around 300 B.C.) to the times of Riemann (1850 A.D.). For a detailed account the reader may refer to Bourbaki [6], Hawkins [16] and Kline [20]. The main aim of the historical notes is to make young readers aware of the fact that mathematical concepts arise out of physical problems and that it can take centuries for a concept to evolve. This section also includes Riemann's example of an integrable function having an infinite number of discontinuities and a proof of the fundamental theorem of calculus due to G. Darboux. The next section of Chapter 1 has a discussion about the drawbacks of the Riemann integral, including the example due to Vito Volterra (1881) of a differentiable function / : [0,1] —> R whose derivative function is bounded but is not Riemann integrable. These considerations made mathematicians look for an extension of the Riemann integral and eventually led to the construction of the Lebesgue integral. Chapter 2 discusses two significantly different approaches for extending the notion of the Riemann integral. The one due to H. Lebesgue is sketched Preface xin in this chapter and is discussed in detail in the rest of the book. The second is due to P.J. Daniel [10] and F. Riesz [32], an outline of which is given. These discussions motivate the reader to consider an extension of the length function from the class of intervals to a larger class of subsets of EL Chapters 3, 4 and 5 form the core of the subject: extension of measures and the construction of the integral in the general setting with the Lebesgue measure and the Lebesgue integral being the motivating example. The pro• cess of extension of additive set functions (known as the Caratheodory ex• tension theory) is discussed in the abstract setting in Chapter 3. This chapter also includes a result due to S.M. Ulam [41] which rules out the possibility of extending (in a meaningful way) the length function to all subsets of R, under the assumption of the Continuum Hypothesis. The outcomes of the general extension theory, as developed in Chapter 3, are harvested for the particular case of the real line and the length function in Chapter 4. This gives the required extension of the length function, namely, the Lebesgue measure. Special properties of the Lebesgue measure and Lebesgue measurable sets (the collection of sets on which the Lebesgue measure is defined by the extension theory) with respect to the topology and the group structure on the real line are discussed in detail. The final section of the chapter includes a discussion about the impossibility of extending the length function to all subsets of E under the assumption of the Axiom of Choice. In Chapter 5, the construction of the extended notion of integral is discussed. Once again, the motivation comes from the particular case of functions on the real line. Lebesgue's recipe, as outlined in Chapter 2, is carried out for the abstract setting. The particular case gives the required integral, namely, the Lebesgue integral. The space Li[a,fe] of Lebesgue in- tegrable function on an interval [a, b] is shown to include TZ[a, 6], the space of Riemann integrable functions, the Lebesgue integral agreeing with the Riemann integral on lZ[a. b]. Also, it is shown that L\[a.7 b] is the completion of 1Z[a, b] under the Li-metric. The final section of the chapter discusses the relation between the Lebesgue integral and the improper Riemann integral. Chapter 6 gives a complete proof of the fundamental theorem of cal• culus for the Lebesgue integral. (This theorem characterizes the pair of functions F, / such that F is the indefinite integral of /. This removes one of the main drawbacks of Riemann integration.) As applications of the fun• damental theorem of calculus, the chain-rule and integration by substitution for the Lebesgue integral are discussed. The remaining chapters of the book include special topics. Chapter 7 deals with the topic of measure and integration on product spaces, with XIV Preface
Fubini's theorem occupying the central position. The particular case of Lebesgue measure on R2 and its properties are discussed in detail. Chapter 8 starts with extending the concept of integral to complex valued functions. The remaining sections discuss various methods of ana• lyzing the convergence of sequences of measurable functions. The Lp-spaces and discussion of some of their dense subspaces in the special case of the Lebesgue measure space are also included in this chapter. The last section of the chapter includes a brief discussion on the application of Lebesgue integration to Fourier series. Chapter 9 includes a discussion of the Radon-Nikodym theorem. As an application, the change of variable formulas for Lebesgue integration on Rn are derived. In Chapter 10, the additive set functions, which are not necessarily nonnegative or even real-valued, are discussed. The main aim is to prove the Hahn decomposition theorem and the Lebesgue decomposition theorem. As a consequence, an alternative proof of the Radon-Nikodym theorem is given. This chapter also includes a discussion of complex measures. The text has three appendices. Appendix E gives a proof of the sin• gular value decomposition of matrices, needed in sections 7.4 and 9.3. In Appendix F, functions of bounded variation (needed in section 6.1) are discussed. Appendix G includes a discussion of differentiate transforma• tion and a proof of the inverse function theorem, needed in section 9.3. (In the present edition four more appendices, A,B,C and D, have been added.) The text is sprinkled with 200 exercises, most of which either include a hint or are broken into doable steps. Exercises marked with • are needed in later discussions. The sections and exercises marked with * can be omitted on first reading. Some of the results in the text are credited to the discoverer, but no effort is made to trace the origin of each result. In any case, no originality is claimed.
Prerequisites and course plans The text assumes that the reader has undergone a first course in mathe• matical analysis (roughly equivalent to that of first five chapters of Apostol [2]). The text as such can be used for a one-year course. A recipe for a one-semester course (approximately 40 lecture hours and 10 problem discus• sion hours) on Lebesgue measure and integration is given after the preface. Since the text is in a lecture-note style, it is also suitable for an individ• ual self-study program. For such readers, the chart depicting the logical interdependence of the chapters will be useful. Preface xv
Acknowledgments It is difficult to list all the individuals and authors who have influenced and helped me in preparing this text, directly or indirectly. First of all I would like to thank my teacher and doctoral thesis advisor, Prof. K.R. Parthasarathy (Indian Statistical Institute, Delhi), whose lectures at the University of Bombay (Mumbai) and the Indian Statistical Institute (Delhi), clarified many concepts and kindled my interest in the subject. I learned much from his style of teaching and mathematical exposition. Some of the texts which have influenced me in one form or another are Halmos [14], Royden [34], Hewitt and Stromberg [18], Aliprantis and Burkinshaw [1], Friedman [13] and Parthasarathy [28]. I am indebted to the students to whom I have taught this subject over the years for their reactions, remarks, comments and suggestions which have helped in deciding on the style of presentation of the text. It is a pleasure to acknowledge the support and encouragement I received from my friend Prof. S. Kumaresan (University of Bombay) at various stages in the preparation of this text. He also went through the text, weeding out misprints and mistakes. I am also thankful to my friend Dr. S. Purkayastha (Indian Institute of Technology Bombay) for going through the typeset man• uscript and suggesting many improvements. For any shortcoming still left in the text, the author is solely responsible. I thank C.L. Anthony for processing the entire manuscript in I^TfrjX. The hard job of preparing the figures was done by P. Devaraj, I am thankful for his help. Thanks are also due to the Department of Mathematics, IIT Bombay for the use of Computer Lab and photocopying facilities. I would like to thank the Curriculum Development Program of the Indian Institute of Technology Bombay for the financial support to prepare the first version of the manuscript. The technical advice received from the production de• partment of Narosa Publishers in preparing the camera ready copy of the manuscript is acknowledged with thanks. Special thanks are due to my family: my wife Lalita for her help in more ways than one; and my parents for allowing me to choose my career and for their love and encouragement in pursuing the same. It is to them that this book is dedicated. Finally, I would be grateful for critical comments and suggestions for later improvements.
Mumbai, 1997 This page intentionally left blank Preface to the Second Edition
In revising the first edition, I have resisted the temptation of adding more topics to the text. The main aim has been to rectify the defects of the first edition:
• Efforts have been made to remove the typos and correct the mis• matched cross references. I hope there are none now. • In view of the feedback received from students, at many places phrases like 'trivial to verify', 'easy to see', etc have been expanded with explanations. • Sequencing of topics in some of the chapters has been altered to make the development of the subject matter more consistent. • Short notes have been added to give a glimpse of the link between measure theory and probability theory. • More exercises have been added. • Four new appendices have been added.
While preparing the first edition of this book, I was often questioned about the 'utility' of spending my valuable 'research time' on writing a book. The response of the students to the first edition and the reviewers' comments have confirmed my confidence that writing a book is as valuable as doing research. I thank all the reviewers of the first edition for their encouraging remarks. Their constructive criticism has helped me a lot in preparing this edition.
xvn XV111 Preface to the Second Edition
I would like to thank Mr. N. K. Mehra, Narosa Publishers, for agreeing to copublish this edition with the AMS. I take pleasure in offering thanks to Edward G. Dunne, Acquisitions Editor, Book Program AMS, for the help and encouragement received from him. I thank the editorial and the technical support staff of the AMS for their help and cooperation in preparing this edition. Once again, the help received from P. Devraj in revising the figures is greatly appreciated. Thanks are also due to Mr. C.L. Anthony and Clarity Reprographers & Traders for typesetting the manuscript in I^TEK. I would greatly appreciate comments/suggestions from students and teachers about the present edition. I intend to post comments/corrections on the present edition on my homepage at www.math.iitb.ac.in/~ikr/books.html
Mumbai, 2002 Inder K. Rana Recipe for a one semester course and interdependence of the chapters
*Lebesgue measure and integration (40 lectures and 10 problem/discussion hours)
Prologue: Everything Chapter 1: Sections 1.1 and 1.2 (depending upon the background of the students), 1.3 and 1.4 can be left for self study. Chapter 2: Sections 2.1 and 2.2. Chapter 3: Sections 3.1 to 3.3; 3.5 to 3.9; 3.10 and 3.11 (omitting the proofs). Chapter 4: Sections 4.1 to 4.3; 4.4 and 4.5 (omitting the proofs); 4.6. Chapter 5: Sections 5.1 to 5.6; Parts of 5.7 to 5.9 can be included de• pending upon the background of students and the emphasis of the course. Chapter 6: Sections 6.1; 6.2 (omitting proofs); 6.3 (stating the theorem 6.3.6 and giving applications: 6.3.8, 6.3.10 to 6.3.13, 6.3.16 (omitting proof). Chapter 7: Sections 7.1 to 7.4.
xix xx Recipe for a one semester course and interdependence of the chapters
Interdependence of the chapters
Prologue The length function
Chapter 1 Riemann integration
Chapter 2 Recipes for extending the Riemann integral
Chapter 3 Chapter 3 General extension 3.2, 3.6 and 3.11 Chapter 4 theory only The Lebesgue measure on IR and Chapter 5 its properties Chapter 5 5.1 to 5.5 only Integration
Chapter 6 Chapter 7 Fundamental theorem Measure and Integration on product spaces of calculus for the Lebesgue integral Chapter 8 Modes of convergence and Lp-spaces
Chapter 9 Radon-Nikodym theorem and its Chapter 10 applications Chapter 10 Signed measures and 10.1 and 10.2 only complex measures Notations used in the text
The three digit system is used to number the definitions, theorems, propo• sitions, lemmas, exercises, notes and remarks. For example, Theorem 3.2.4 is the 4th numbered statement in section 2 of chapter 3. The symbol • is used to indicate the end of a proof. The symbol A := B or B =: A means that this equality is the definition of A by B. The symbol • before an exercise means that this exercise will be needed in the later discussions. Sections, theorems, propositions, etc., which are marked * can be omitted on first reading. The phrase "the following are equivalent:" means each of the listed state• ment implies the other. For example in Theorem 1.1.4, it means that each of the statements (i), (ii) and (iii) implies the other. The notations and symbols used from logic and elementary analysis are as follows:
implies; gives 7^ does not imply implies and is implied by; if and only if 3 there exists V for all; for every x e A x belongs to A x g A x does not belong to A AcB A is a proper subset of B ACB A is a subset of B V{X) set of all subsets of X
xxi XXII Notations used in the text
0 empty set A\B set of elements of A not in B AxB Cartesian product of A and B n n* Cartesian product of sets X\,... , Xn. inf infimum sup supremum Uu union intersection aA n symmetric difference Ac complement of a set A E closure of a set E dE boundary points of a set E lim sup limit superior; upper limit lim inf limit inferior; lower limit n—>oo
> : / is a function from X into Y and f(x) — y.
the set of natural numbers the set of integers the set of rational numbers the set of real numbers the set of extended real numbers the set of complex numbers n-dimensional Euclidean space absolute value of x
(a, 6), (a, b], [a, b) [a, b], (-00, oc), j ; .^^ ^ R_ (—00, a), (—00, aj, (a, oo), [a, oo) J
th {«n}n>i • sequence with n term an. (X, d) : a metric space.
For the list of other symbols used in the text, see the symbol index given at the end of the text. References
[1] Aliprantis, CD. and Burkinshaw, O. Principles of Real Analysis (3rd Edition). Academic Press, Inc. New York, 1998.
[2] Apostol, T.M. Mathematical Analysis. Narosa Publishing House, New Delhi (India), 1995.
[3] Bartle, Robert G. A Modern Theory of Integration, Graduate Studies in Mathemaics, Volume 32, American Mathematicsl Society,Providence, RI, 2001.
[4] Bhatia, Rajendra Fourier Series. Hindustan Book Agency, New Delhi (India), 1993.
[5] Billingsley, Patrick Probability and Measure. 3rd Edition, John Wiley and Sons, New York, 1995.
[6] Bourbaki, N. Integration, Chap. V. Actualites Sci. Indust. 1244. Hermann, Paris, 1956.
[7] Carslaw, H.S. Introduction to the Theory of Fourier's Series and In• tegrals. Dover Publications, New York, 1952.
[8] Caratheodory, C. Vorlesungen iiber Reelle Funktionen. Leipzig, Teub- ner, and Berlin, 1918.
409 410 References
[9] Caratheodory, C. Algebraic Theory of Measure and Integration. Chelse Publishing Company, New York, 1963 (Originally published in 1956).
[10] Daniell, P.J. A general form of integral. Ann. of Math. (2)19 (1919), 279-294.
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[12] Fraenkel, A. A.Abstract Set Theory, Fourth Edition, North-Holland, Amsterdam, 1976.
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[14] Halmos, P.R. Measure Theory. Van Nostrand, Princeton, 1950.
[15] Halmos, P.R. Naive Set Theory. Van Nostrand, Princeton, 1960.
[16] Hawkins, T.G. Lebesgue's Theory of Integration: Its Origins and De• velopment. Chalsea, New York, 1979.
[17] Hewitt, E. and Ross, K.A. Abstract Harmonic Analysis, Vol.1. Springer- Verlag, Heidelberg, 1963.
[18] Hewitt, E. and Stromberg, K. Real and Abstract Analysis. Springer- Verlag, Heidelberg, 1969.
[19] Kakutani, S. and Oxtoby, J.C. A non-separable translation invariant extension of the Lebesgue measure space. Ann.of Math. (2) 52 (1950), 580-590.
[20] Kline, M. Mathematical Thoughts from Ancient to Modern Times. Oxford University Press, Oxford, 1972.
[21] Kolmogorov, A.N. Foundations of Probability Theory. Chelsea Pub• lishing Company, New York, 1950.
[22] Korner, T.W. Fourier Analysis. Cambridge University Press, London, 1989. References 411
[23] Lebesgue, H. Integrale, longueur, aire. Ann. Math. Pura. Appl. (3) 7 (1902), 231-259.
[24] Luxemburg, W.A.J. Arzela's dominated convergence theorem for the Riemann integral Amer. Math. Monthly 78 (1971), 970-979.
[25] McLeod, Robert M., The Generalized Riemann Integral, Carus Mono• graph, No.20, Mathemaical Associaiton of America, Washington, 1980.
[26] Munkres, James E. Topology, 2nd Edition, Prentice Hall, Englewood Cliffs, NJ, 1999.
[27] Natanson, LP. Theory of Functions of a Real Variable. Frederick Un- gar Publishing Co., New York, 1941/1955.
[28] Parthasarathy, K.R. Introduction to Probability and Measure. Macmillan Company of India Ltd., Delhi, 1977.
[29] Parthasarathy, K. R. Probablity Measures on Metric Spaces, Academic Press, New York, 1967.
[30] Rana, Inder K. From Numbers to Analysis, World Scientific Press, Singapore, 1998.
[31] Riesz, F. Sur quelques points de la theorie des fonctions sommables. Comp. Rend. Acad. Sci. Paris 154 (1912), 641-643.
[32] Riesz, F. Sur Vintegrale de Lebesgue. Acta Math. 42 (1920), 191-205.
[33] Riesz, F. and Sz.-Nagy, B. Functional Analysis. Fredrick Ungar Pub• lishing Co., New York, 1955.
[34] Royden, H.L. Real Analysis (3rd Edition). Macmillan, New York, 1963.
[35] Saks, S. Theory of the Integral. Monografje Matematyczne Vol. 7, Warszawa, 1937.
[36] Serrin, J. and Varberg, D.E. A general chain-rule for derivatives and the change of variable formula for the Lebesgue integral. Amer. Math. Monthly 76 (1962), 514-520. 412 References
[37] Solovay, R. A model of set theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2) 92 (1970), 1-56.
[38] Srivastava, S. M. Borel Sets, Springer-Verlag, Heidelberg, 1998.
[39] Stone, M.H. Notes on integration, I-IV. Proc. Natl. Acad. Sci. U.S. 34 (1948), 336-342, 447-455, 483-490; 35 (1949), 50-58.
[40] Titchmarch, E.C. The Theory of Functions. Oxford University Press, Oxford, 1939 (revised 1952).
[41] Ulam, S.M. Zur Masstheorie in der allgemeinen Mengenlehre. Fund. Math. 16 (1930), 141-150.
[42] Zygmund, A. Trigonometric Series, 2 Vols. Cambridge University Press, London, 1959. Index
2 Ho, 112 -subsets of M , 211 A®B, 210 bounded a.e., 125 - convergence theorem, 151 absolutely continuous - linear functional, 302, 375 - complex measure, 366 - variation, 397 - function, 176 - measure, 311 C, 243 - signed measure, 356, 359 C(R), 162 aleph-nought, 391 C[a,b], 41, 160 algebra, 55 C°°[a,6], 167 - generated, 57 C°°-function, 286 almost everywhere, 125 C°°(1R) , 16 4 almost uniformly convergent, 250 C°°(U), 286 analytic set, 112 C~(t/), 286 antiderivative, 30 1 C -mapping, 403 approximate identity, 291 Co (M"), 295 Archemedes, 30 C (R), 162 arrangement, 387 C C (Rn), 280, 295 Arzela's theorem, 40, 157 c C n E, 56 c, 112
BR, 95 XA(x), 30 3*1,211 Cantor sets, 23, 24 £x,8 2 Cantor's ternary set, 25 Baire, Rene, 44 Carat heodory, Constant in, 44 Banach cardinal number, 391 - algebra with identity, 290 cardinality - algebra, commutative, 290 - of a set, 391 - lattice, 365 - of the continuum, 112, 391 - spaces, 266 Cartesian product, 389 Bernoulli, Daniel, 31 Cauchy, Augustein-Louis, 31 Bessel's inequality, 298, 308 Cauchy in measure, 259 binomial distribution, 69 Cauchy-Schwartz inequality, 264, 296 Borel, Emile, 44 chain rule, 204 Borel change of variable for Riemann integration, - measurable function, 227 200, 207 - subsets, 95, 102 change of variable formula
413 414 Index
- abstract, 332 Dini derivatives, 180 - linear, 334 Dini, Ulisse, 37 - nonlinear, 338 direct substitution, 36 charactristic function, 30 Dirichlet function, 32 Chebyshev's inequality, 146, 270 Dirichlet, Peter Gustev Lejeune, 32 choice function, 390 discrete, closed subspace, 298 - measure, 68 complete measure space, 92 - probability measure, 68 completion of a measure space, 92 distribution, complex measure, 365 - binomial, 68 complex numbers, 243 - discrete probability, 68 conditional expectation, 305, 321 - function, 64, 115 conjugate, 374 - function, probability, 115 continuity from above or below, 66 - of the measurable function, 142 continuum hypothesis, 392 - Poisson, 68 convergence, - uniform, 68 - almost everywhere, 248 dominated convergence theorem, an exten• - almost uniform, 250 sion, 274 - pointwise, 248
- uniform, 248 Ex, 215 convergence in EV, 215 - Lp, 267 Egoroff's theorem, 249 - measure, 255 equicontinuous, 271 - pth mean, 267 equipotent, 391 - probability, 261 equivalent, 158 convergent to essential supremum, 291 - +oo, 386 essentially bounded, 291 - -oo, 386 Euclid, 30 convolution, 283 Euler's identity, 310 coordinate functions, 401 Euler, Leonhard, 30 countable, example, - set, 391 - Riemann's, 32 - subadditivity, 4 - Vitali's, 113 countably - Volterra's, 37 - additive, 59 extended, - subadditive, 59 - integration by parts, 200 counting measure, 225, 312 - real numbers, 1, 386 critical values, 98 extension, cylinder set, 58 - of a measure, 60, 83, 88 cylindrical coordinates transformation, 344 - of the dominated convergence theorem, 274 (Dfi)(x), CD/z)(s), 327 (£>+/)(c), (D-/)(c), 180 U*9), 283 (D+/)(c), (£>-/)(c), 180 , 296 ^W, 320 f~g, 43, 158 d[i f-X(C), 57 Dfj,(x), 324 fn -^ /, 255 d'Alembert, Jean, 31 Darboux, Gaston, 34 fn -^ /, 248 Darboux's theorem, 18 fn -±> /, 248 Denjoy integral, 172 fn ^ /, 248 derivative of a measure, 324 fn ^ /, 250 Descartes, Rene, 31 HC), 57 det(T), 334 Fatou's lemma, 140 determinant, 334 finite differentiable, 324, 401 - additivity property, 1 differential, 401 - set, 391 Index 415
- signed measure, 346 Hilbert space, 297 finitely additive, 59 Holder's inequality, 263 Fourier - coefficients, 32, 307 X, 1, 55 - series, 32, 307 J, 55 Fourier, Joseph, 31 Jo, 96 Fubini's theorem, 189, 221, 222 Id, 82 function, Ir, 82 - absolutely continuous, 176 Im(/), 243 -C°°, 286 / fdfj., 123, 244 - characteristic, 30 J sdfi, 119 - choice, 390 fZf(x)dx, 10 - Dirichlet's, 32 fz_fdv, 125, 145 - generalized step, 48 b - imaginary part of, 244 fa f(x)dx, 10 - indefinite integral of, 175 Sjf{x)dx , 10 - indicator, 30 improper Riemann integral, 168, 169 - infinitely differentiable, 163 indefinite integral, 175 - integrable, 143, 244, 358 indicator function, 30 - Lebesgue singular, 181 infinitely differentiable functions, 164 - length, 1 inner product, 296 - Lipschitz, 177 inner regular, 103 - locally integrable, 286 integrable, 143, 244, 358 - measurable, 135 integral, 123, 144, 244 - negative part of , 49 - lower, 10 - nonnegative measurable, 118, 123 - of nonnegative simple measurable func• - nonnegative simple measurable, 118 tion, 119 - of bounded variation, 397 - over E , 125 - popcorn, 26 - upper, 10 - positive part of, 49, 135 integration - real part of, 244 - by parts, 36, 199 - Riemann integrable, 10 - by substitution, 206 - Riemann integral of, 10 - of radial functions, 234 - simple, 48 intervals, - simple measurable, 135 - with dyadic endpoints, 82 - step, 47 - with rational endpoints, 82 - support of, 162 inverse - vanishing at infinity, 283 - function theorem, 405 - with compact support, 280 - substitution, 36 fundamental theorem of calculus, 34, 191, 195, 197 Jacobian, 405 Jordan, Camille, 44 gamma function, 173 Jordan decomposition theorem, 350 gauge integral, 172 Jordan's theorem, 398 generalized Riemann integral, 172 generated, kernel, 303 - algebra, 57 Kurzweil-Henstock integral, 172 - monotone class, 86 - cr-algebra, 81 L, 135 graph of the function, 219 L+, 123 lo, 48 Haar measure, 108 L+,118 Hahn LJ(X,5,M), 244 - decomposition, 350, 373 L[oc(Rn) , 286 - theorem, 349 Li(X,S,/i), 144 Hankel, Hermann, 36 Li[a,6], 154 Heine-Borel theorem, 108 Li-metric, 159 416 Index
Li(E), 154 /x+, 351 LJ(X,
Mb(X,S), 362 nonnegative simple measurable function, 118 \x JL v, 319 norm, 159, 266 fi V v, 365 - induced by the inner product, 297 \i A i/, 365 - of a bounded linear functional, 376 /xT~\ 332 - of a complex measure, 371 /i-null set, 347 - of a partition, 6 Index 417
- of a signed measure, 363 (M,£F,/xF), 314 normed linear space, 266 R, 1 null Re(/), 243 - set, 22 .R-integrable, 17 - subset, 92 n[a,b], 18 Radon, Johann, 44 ft, 111 Radon-Nikodym derivative, 320 w(/,J), 20 Radon-Nikodym theorem, - for complex measures, 367 "(/,*), 20 - for finite measures, 354 open intervals, 82, 96 - for measures, 319 orthogonal, 297 - for signed measures, 357 - complement, 298 random variable, 261 oscillation of a function, real part, 244 - at a point, 20 refinement of a partition, 7 - in an interval, 20 regular outer measure, 73 - measure, 324 - induced, 71 - partition, 14, 15 - Lebesgue, 95 regularity of A 2, 229 outer regular, 102 R regularization of a function, 287 outer regularity of L A, 102 representation, standard, 118 Riemann, Bernhard, 32 th p norm of /, 262 Riemann parallelogram identity, 297 - integrable, 10 Parseval's identity, 309 - integral, 10 partial sum of the Fourier series, 307 - sum, 17 partition, Riemann-Lebesgue lemma, 163 -6 Riemann's example, 32 - measurable, 129 Riesz representation, 303 - norm of, 6 Riesz representation theorem, 377 - refinement of, 7 Riesz theorem, 257 - regular, 14 Riesz, Friedrich, 44 Perron integral, 172 Riesz-Fischer theorem, 265, 308 pointwise, 248 Poisson distribution, 69 S(PJ), 17 polar, S^, 298 - coordinate, 341 s± Vs2, 120 - coordinate transformation, 340 si As2, 120 - representation, 372 5(C), 81 popcorn function, 26 S*, 76 positive part, Saks' theorem, 97 - of a function, 16, 49, 135 section of E at x or y, 215 - of a signed measure, 351 semi-algebra, 54 positive set, 347 set function, 59 power set, 55 - countably additive, 59 probability, 94 - countably subadditive, 59 - distribution function, 115 - finite, 84 - measure, discrete, 68 - finitely additive, 59 - space, 94 - induced, 62 product, - monotone, 59 - measure, 238 - cr-finite, 84 - measure space, 213, 238 sigma algebra, 80 - of measures (i and v, 213 cr-algebra, - cr-algebra, 210, 212 - generated, 81 projection theorem, 301 - monotone class technique, 88 pseudo-metric, 43 - monotone class theorem, 87 Pythagoras identity, 298 - of Borel subsets of R, 95 418 Index
- of Borel subsets of X, 82 - monotone convergence, 127 - product, 210 - Riesz representation, 377 - technique, 82 - Riesz-Fischer, 159, 308 cr-fmite, - Saks', 97 - set function, 84 - cr-algebra monotone class, 87 - signed measure, 346 - Steinhaus, 104 cr-set, 57 - Ulam's, 61 signed measure, 345 - von Neumann, 315 simple - Vitali covering, 108 - function, 48, 279 topological, - function technique, 152 - group, 108 - measurable function, 135 - vector space, 261 - nonnegative measurable function, 118 total variation, 397 singular - of a complex measure, 368 - measure, 319 - of a signed measure, 351 - value decomposition, 395 totally finite, 84 - values, 396 transition, smaller, 365 - measure, 229 smoothing of a function, 281 - probability, 229 space, translation invariance, 4, 230 - Banach, 266 triangle inequality, 266 - Hilbert, 297 truncation sequence, 141 - inner product, 297 - Lebesgue measure, 95 Ulam's theorem, 61 - measurable, 92 ultrafilter, 70 - measure, 92 uncountable, 391 - normed, 266 uniform distribution, 69 - probability, 94 uniformly, - product measure, 213 - absolutely continuous, 271 spherical coordinates transformation, 344 - integrable, 275 standard representation, 118 upper, Steinhaus theorem, 104 - derivative, 327 step function, 47, 50, 251 - integral, 10 subspace, 298 - left derivative, 180 - closed, 298 - left limit, 180 sum, - right derivative, 180 - lower or upper, 7 - right limit, 180 - Riemann, 17 - sum, 7 supp (/), 162 - variation, 351 support, 162 Urysohn's lemma, 280, 393 symmetric moving average, 281 V?(/), 397 Theorem, Vj(P,/), 397 - Arzela's, 40, 157 - bounded convergence, 151 variation, 397 - Darboux's, 18 - on R, 198 - Egoroff's, 249 Vitali - Pubini, 189, 221, 222, 239 - cover, 108 - Fundamental theorem of calculus, 34 - covering theorem, 108 - Hahn decomposition, 349 Vitali's example, 113 - Heine-Borel, 108 Volterra's example, 37 - inverse function, 405 Volterra, Vito, 37 - Jordan, 398 von Neumann theorem, 315 - Jordan decomposition, 350 Weirstrass, Karl, 36 - Lebesgue - Young, 179 - Lebesgue's dominated convergence, 148 (X,5,/Z), 92 - Luzin's, 254 Index of notations
Prologue
Set of real numbers, 1 Extended real numbers, 1 1 The collection of all intervals, [0, +00] The set {x GM*|x>0}, 1 Empty set, 1
Chapter 1 ll^ll Norm of a partition P, 6 L(PJ) Lower sum of / with respect to P, 6 U(PJ) Upper sum of / with respect to P, 7 P1UP2 Common refinement of P\ and P2, 7
Lower integral, 10 J a
I f(x)d:X Riemann integral of / over [a, 6], 10 J a
I f(x)d Upper integral, 10 J a
Regular partition of an interval 14 Positive part of a function, 16 Negative part of a function, 16
419 420 Index of notations
S(PJ) Riemann sum of / with respect to P, 17 K[a, b] Set of Riemann integrable functions on [a, 6], 18 Oscillation of / in the interval J, 20 Oscillation of / at x, 20 Characteristic function of the set A, 30 Chapter 2
Ln : Collection of simple functions on R, 48
Chapter 3 Hi) The algebra generated by intervals, 52 i The collection of left-open, right-closed intervals, 55 V(X) Power set of X; the collection of all subsets of X, 55 cnE Subsets of E which are elements of C, 56 r\E) The set {x\f(x) G £}, 57 HC) The algebra generated by C, 57 Set function induced by the function F, 62 M* Outer measure induced by /i, 71
Bx The a-algebra of Borel subsets of a topological space X, 82 BR The cr-algebra of Borel subsets of R 82 Open intervals with rational end points, 82 id Subintervals of [0,1] with dyadic end points, 82 M(C) Monotone class generated by C, 86 CF The cr-algebra of //^-measurable sets, 88 HF Lebesgue-Stieltjes measure induced by F, 89 (X,S ) Measurable space, 92 Measure space, 92 (X,S,/Z ) The completion of a measure space (X, 5, //), 92
Chapter 4
A* Lebesgue outer measure, 95 C cr-algebra of Lebesgue measurable subsets of R, 95 A The Lebesgue measure on R, 95
X0 The collection of all open intervals in R, 96 diameter(F) The diameter of a subset E of R, 96 fix) The derivative of / at x, 96 A + x The set {y + x\y G A}, 101 xE The set {xy\y G E}, 103 Cardinality of the continuum, 110 Index of notations 421 ft : The first uncountable ordinal, 111 : Aleph nought, the cardinality of the set N, 112
Chapter 5 U The collection of nonnegative simple measurable functions, 118
sdji Integral of a function sGLj" with respect to /i, 119 / S\ V 52 Maximum of the functions s\ and 52, 120
51 A 52 Minimum of the functions s\ and 52, 120 L+ The class of nonnegative measurable functions, 123 Integral of/GL+, 123 / fdji
P a.e. x(n) on 7) , , ,. _ , \ ^ > : rn i holdn s tor almos xt every x G Y P a.e. (ii)x G Yr J J with respect to /x, 125
/ fdfi : Integral of / over E with respect to /i, 125 JE
L : The class of measurable functions, 135
L0 : The class of simple measurable functions, 135
i(X,5,/x) 1 t The space of yu-integrable functions, 144 i(X), Li(/x) J '
Li(E) The space of integrable functions on £", 154 Li[a,b] The space of integrable functions on [a, 6], 154 The Li-norm of /, 158 C[a, b] The space of continuous functions on [a, 6], 160 supp(/) Support of a function, 162 The space of continuous functions on R with a compact support, 162 c° The space of infinitely differentiable functions on K, 164 C°°[a,b] The space of infinitely differentiable functions on [a, 6], 167 Cf The space of functions in C°°(IR) with compact support, 168 POO / f{x)dx Improper Riemann integral of / over [a, oo), 168 J a 422 Index of notations
Chapter 6
Variation of / over the interval [a, 6], 176 liminf $(x) Lower right limit of $ at c, 180 hie limsup$(x) Upper right limit of <& at c, 180 hie liminf $(x) Lower left limit of $ at c, 180 limsup$(x) Upper left limit of $ at c, 180 h^c
(D+f)(c) Lower right derivative of / at c, 180 (D+f)(c) Upper right derivative of / at c, 180 (D-f)(c) Lower left derivative of / at c, 180 (D~f)(c) Upper left derivative of / at c, 180 Variation of F on R, 198
Chapter 7
A ® B : Product of the cr-algebra A with B, 211 2 BR2 : The cr-algebra of Borel subsets of R , 211 (X x Y, A ® B, /J, x v) : The product measure space, 213
Ex Section of E at x, 215 Section of E at y, 215 Lebesgue measurable subsets of R2, 229 Lebesgue measure on R2, 229 f2 The set {/x J\I,J el}, 229 detT Determinant of T, 232
TT -X"i> ($0 A, TT /^ J : Product of a finite number \i=\ 2=1 i=i of measure spaces, 237
Lebesgue measurable subsets of Rn, 239 Afl£n Lebesgue measure on Rn, 239 B^n The cr-algebra of Borel subsets of Rn, 239 S(x,r) The open ball in Rn with center at x and radius r, 240 Index of notations 423
Chapter 8
C Field of complex numbers, 243 Re(/) Real part of a complex-valued function /, 243 Im(/) Imaginary part of a complex-valued function /, 244 L\(X,S,fi) Real-valued /^-integrable functions, 244 Li(X,S,n) Complex-valued /x-integrable functions, 244 > Jn J fn converges to / pointwise, 248 > Jn J fn converges to / almost everywhere, 248 fn^f fn converges to / uniformly, 248 a.u. p / n y J fn converges to / almost uniformly, 250 P m j, Jn • / fn converges to / in measure, 255 M(X) The set of all measurable functions on X, 260
L (X,S,n) p : The space of pth-power integrable functions of /, 262 Lp{n)
th II/IIP p norm of /, 262 fh The function fh(x) := f(x + /i)Vx, 281 f*9 Convolution of / with g, 283 n Llfc(Rn) Space of locally integrable functions on M , 286 C°°{U) The set of infinitely differentiable functions on £/, 287 C?{U) The set {/ G C°°(U) | supp(/) is compact}, 287 5(0,1) Closure of the ball 5(0,1), 287 Loo(X,5,/i) Space of essentially bounded functions, 291 ll/lloo Essential supremum of /, 292 n n C0(R ) The set of continuous functions on R vanishing at infinity, 295 L2(X,S,fx) Space of square integrable functions, 296 ll/lh I/2-norm of /, 296 (f,9) Inner product of /, g G L2, 296 X 5 The set {/ G L2(X,S,M)|
Chapter 9
v is absolutely continuous with respect to //, 311
(R,BF,/zF) Completion of the measure space (IR, Z3R, /ip), 314 /i J_ z/ fi is singular with respect to v, 319 Radon-Nikodym derivative of v with respect to //, 320 (£>/*)(*) Derivative of /i at x, 324 (Dfi)(x) Upper derivative of \x at x, 327 (D»)(x) Lower derivative of /i at x, 327 424 Index of notations
The measure induced by T, 332 JT(x) : Jacobian of T at x, 336 Chapter 10
Upper variation of a signed measure /x, 351 A* Lower variation of a signed measure /x, 351 IA*I Total variation of a signed measure /x, 351 Absolute continuity of a signed measure v with respect to a measure /x, 356 Integral of / with respect to a signed measure /x, 358 v < // v absolutely continuous with respect to a signed measure /x, 359
-M6(X,
X equipotent to F, 391 card(A) Cardinality, cardinal number of A, 391 No Cardinality of N, 391 c Cardinality of the continuum, 391 2 card(x) Cardinality of the power set of X, 391 A* Transpose of a matrix, 395 V*(PJ) Variation of / over [a, b] with respect to a partition P, 397 va\f) Variation of / over [a, 6], 397 Differential of a differentiate mapping T, 401 (dT)(a) jth partial derivative of Ti at x, 401 Jacobian of T at x, 405 JT{X) Titles in This Series
45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, 2002 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 TITLES IN THIS SERIES
5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993