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Homeomorphisms in Analysis http://dx.doi.org/10.1090/surv/054 ERRATUM TO "HOMEOMORPHISMS IN ANALYSIS" BY CASPER GOFFMAN, TOGO NISHIURA, AND DANIEL WATERMAN An omission appears in a display on page 113. The display on line 8 from the bottom of the page should appear as follows |/(x)| = lim M*;/)| < supllSnCOlU < ||/||u. n-»oo n Selected Titles in This Series 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors, The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986 20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986 (Continued in the back of this publication) Mathematical Surveys and Monographs Volume 54 Homeomorphisms in Analysis Casper Goffman Togo Nishiura Daniel Waterman ^HEM^y American Mathematical Society Editorial Board Howard A. Masur Michael Renardy Tudor Stefan Ratiu, Chair 1991 Mathematics Subject Classification. Primary 26-02, 28-02, 42-02, 54-02; Secondary 30-02, 40-02, 49-02. ABSTRACT. The interplay of two of the main branches of Mathematics, topology and real analysis, is presented. In the past, homeomorphisms have appeared in books on analysis in incidental or casual fashion. The presentation in this book looks deeply at the effect that homeomorphisms have, from the point of view of analysis, on Lebesgue measurability, Baire classes of functions, the derivative function, Cn and C°° functions, the Blumberg theorem, bounded variation in the sense of Cesari, and various theorems on Fourier series and generalized variation of functions. Unified developments of many results discovered over the past 50 years are given in book form for the first time. Among them are the Maximoff theorem on the characterization of the derivative function, various results on improvements of convergence or preservation of convergence of Fourier series by change of variables, and approximations of one-to-one mappings in higher dimensions by homeomorphisms. Included here is the von Neumann theorem on homeomorphisms of Lebesgue measure. Library of Congress Cataloging-in-Publication Data Goffman, Casper, 1913- Homeomorphisms in analysis / Casper Goffman, Togo Nishiura, Daniel Waterman. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 54) Includes bibliographical references (p. - ) and index. ISBN 0-8218-0614-9 (alk. paper) 1. Homeomorphisms. 2. Mathematical analysis. I. Nishiura, Togo, 1931- . II. Waterman, Daniel. III. Title. IV. Series: Mathematical surveys and monographs ; no. 54. TA355.T47 1998 620.3—dc21 97-25854 CIP 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 publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. 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 homepage at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97 DEDICATED TO THE MEMORIES OF OUR MENTORS HENRY BLUMBERG, LAMBERTO CESARI, ANTONI ZYGMUND Contents Preface xi The one dimensional case xi Mappings and measures on Rn xii Fourier series xiii Part 1. The One Dimensional Case 1 Chapter 1. Subsets of R 3 1.1. Equivalence classes 3 1.2. Lebesgue equivalence of sets 4 1.3. Density topology 5 1.4. The Zahorski classes 10 Chapter 2. Baire Class 1 13 2.1. Characterization 13 2.2. Absolutely measurable functions 15 2.3. Example 19 Chapter 3. Differentiability Classes 21 3.1. Continuous functions of bounded variation 21 3.2. Continuously differentiable functions 27 3.3. The class Cn [0,1] 30 3.4. Remarks 38 Chapter 4. The Derivative Function 41 4.1. Properties of derivatives 41 4.2. Characterization of the derivative 45 4.3. Proof of Maximoff's theorem 47 4.4. Approximate derivatives 53 4.5. Remarks 56 Part 2. Mappings and Measures on Rn 59 Chapter 5. Bi-Lipschitzian Homeomorphisms 61 5.1. Lebesgue measurability 61 5.2. Length of nonparametric curves 63 5.3. Nonparametric area 68 5.4. Invariance under self-homeomorphisms 71 5.5. Invariance of approximately continuous functions 72 5.6. Remarks 74 viii CONTENTS Chapter 6. Approximation by Homeomorphisms 77 6.1. Background 77 6.2. Approximations by homeomorphisms of one-to-one maps 78 6.3. Extensions of homeomorphisms 80 6.4. Measurable one-to-one maps 84 Chapter 7. Measures on Rn 89 7.1. Preliminaries 89 7.2. The one variable case 91 7.3. Constructions of deformations 91 7.4. Deformation theorem 96 7.5. Remarks 97 Chapter 8. Blumberg's Theorem 99 8.1. Blumberg's theorem for metric spaces 99 8.2. Non-Blumberg Baire spaces 103 8.3. Homeomorphism analogues 104 Part 3. Fourier Series 109 Chapter 9. Improving the Behavior of Fourier Series 111 9.1. Preliminaries 111 9.2. Uniform convergence 117 9.3. Conjugate functions and the Pal-Bohr theorem 120 9.4. Absolute convergence 124 Chapter 10. Preservation of Convergence of Fourier Series 131 10.1. Tests for pointwise and uniform convergence 131 10.2. Fourier series of regulated functions 138 10.3. Uniform convergence of Fourier series 149 Chapter 11. Fourier Series of Integrable Functions 159 11.1. Absolutely measurable functions 159 11.2. Convergence of Fourier series after change of variable 164 11.3. Functions of generalized bounded variation 167 11.4. Preservation of the order of magnitude of Fourier coefficients 178 Appendix A. Supplementary Material 187 Sets, Functions and Measures 187 A.l. Baire, Borel and Lebesgue 187 A.2. Lipschitzian functions 189 A.3. Bounded variation 192 Approximate Continuity 195 A.4. Density topology 195 A.5. Approximately continuous maps into metric spaces 197 Hausdorff Measure and Packing 198 A.6. Hausdorff dimension 198 A.7. Hausdorff packing 199 Nonparametric Length and Area 203 A.8. Nonparametric length 203 A.9. Schwarz's example 203 CONTENTS ix A. 10. Lebesgue's lower semicontinuous area 204 A. 11. Distribution derivatives for one real variable 205 Bibliography 207 Index 213 Preface As the title of the book indicates, there is an interesting role played by homeomor- phisms in analysis. The maturing of topology in the beginning of the 20th Century changed the outlook of analysis in many ways. Earlier, the notions of curve and sur­ face were intuitively obvious to mathematicians.
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