Real Analysis

505 Pages·2002·13.744 MB·English

by  Emmanuele DiBenedetto (auth.)

Birkhauser Advanced Texts
Basler Lehrbiicher
Edited by
Herbert Amann, University of Zurich
Steven G. Krantz, Washington University
Shrawan Kumar, University of North Carolina at Chapel Hill
Emmanuele DiBenedetto
Real Anal ysis
Springer Science+Business Media, LLC
Emmanuele DiBenedetto
Department of Mathematics
Vanderbilt Vniversity
Nashvilie, TN 37240
V.S.A.
Library of Congress Cataloging-in-Publication Data
OiBenedetto, Emmanuele.
Real Analysis / Emmanuele OiBenedetto.
p. cm. - (Birkhăuser advanced texts)
Includes bibliographical references and index.
ISBN 978-1-4612-6620-4 ISBN 978-1-4612-0117-5 (eBook)
DOI 10.1007/978-1-4612-0117-5
1. Mathematical analysis. I. Title. II. Series.
QA300.046 2001
515-dc21 2001052752
CIP
AMS Classification Cades: 03E04, 03EIO, 03E20, 03E25, 26A03, 26A09, 26A12. 26A15, 26A16,
26A21. 26A27, 26A30, 26A42, 26A45, 26A46, 26A48, 26A51, 26805, 26815, 26820, 26B25,
26830.26835, 26B4O. 26EI0. 28A05,28AIO, 28A12, 28A15, 28A20. 28A25,28A33. 28A35,
28A50. 28A 75,28A 78,31B05, 31B 10, 35C15, 35E05. 4OA05,40AlO, 41A 10,42825,42835,
46A03, 46A22, 46A30, 46A32, 46B03, 46B07, 46B 10, 46825, 46B45, 46C05, 46C 15,46E05.
46EIO, 46E15, 46E35. 46F05, 46FIO, 54A05, 54AIO, 54A20, 54A25, 54B05, 54BIO, 54B15,
54C05. 54C30, 54D05, 54010. 54030, 54045, 54060, 54065, 54E35, 54E40, 54E45, 54E50,
54E52
Printed on acid-ti-ee paper
© 2002 Springer Science+Business Media New York
Originally published by Birkhlluser 80ston in 2002
Softcover reprint ofthe hardcover 1s t edition 2002
AII rights reserved. This work may not be translated or copied in whole or in part without the written
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ISBN 978-1-4612-6620-4 SPIN 10798100
Reformatted from the author's files by John Spiegelman, Philadelphia, PA.
987 6 5 4 3 2 I
Contents
Preface xv
Acknowledgments xxiii
Preliminaries 1
Countable sets
2 The Cantor set 2
3 Cardinality.. 4
3.1 Some examples 5
4 Cardinality of some infinite Cartesian products 6
5 Orderings, the maximal principle, and the axiom of choice 8
6 Well-ordering....... 9
6.1 The first uncountable 11
Problems and Complements II
I Topologies and Metric Spaces 17
Topological spaces . . . . . . 17
1.1 Hausdorff and normal spaces . 19
2 Urysohn's lemma ......... . 19
3 The Tietze extension theorem 21
4 Bases, axioms of countability, and product topologies. 22
4.1 Product topologies . . . . . . . . . . . . 24
5 Compact topological spaces . . . . . . . . . . 25
5.1 Sequentially compact topological spaces. 26
vi Contents
6 Compact subsets of ]RN .... .. . . . 27
7 Continuous functions on countably compact spaces 29
8 Products of compact spaces 30
9 Vector spaces ... . . . . . . 31
9.1 Convex sets . . . ... . 33
9.2 Linear maps and isomorphisms 33
10 Topological vector spaces .. 34
10.1 Boundedness and continuity .. 35
11 Linear functionals . . . . . . 36
12 Finite-dimensional topological vector spaces 36
12.1 Locally compact spaces ...... . 37
13 Metric spaces . . . .. .... .. 38
13.1 Separation and axioms of countability 39
13.2 Equivalent metrics 40
13.3 Pseudometrics .... 40
14 Metric vector spaces . . 41
14.1 Maps between metric spaces 42
15 Spaces of continuous functions. 43
15.1 Spaces of continuously differentiable functions 44
16 On the structure of a complete metric space 44
17 Compact and totally bounded metric spaces 46
17.1 Precompact subsets of X 48
Problems and Complements 49
II Measuring Sets 65
Partitioning open subsets of]RN .. ..... . 65
2 Limits of sets, characteristic functions, and a -algebras 67
3 Measures . . . .. .. .. . . 68
3.1 Finite, a-finite, and complete measures 71
3.2 Some examples . ... . 71
4 Outer measures and sequential coverings 72
4.1 The Lebesgue outer measure in ]RN 73
4.2 The Lebesgue-Stieltjes outer measure 73
5 The Hausdorff outer measure in]RN . . . . 74
6 Constructing measures from outer measures . 76
7 The Lebesgue-Stieltjes measure on ]R 79
7.1 Borel measures ... ... . 80
8 The Hausdorff measure on]RN . . . . . 80
9 Extending measures from semialgebras to a -algebras . 82
9.1 On the Lebesgue-Stieltjes and Hausdorff measures 84
10 Necessary and sufficient conditions for measurability 84
II More on extensions from semi algebras to a-algebras .. 86
12 The Lebesgue measure of sets in]RN . . ..... 88
12.1 A necessary and sufficient condition of measurability 88
13 A nonmeasurable set ... .... .... .. .. 90
Contents vii
14 Borel sets, measurable sets, and incomplete measures . . . 91
14.1 A continuous increasing function f : [0, 1] --+ [0, 1] 91
14.2 On the preimage of a measurable set. 93
14.3 Proof of Propositions 14.1 and 14.2 ... . 94
15 More on Borel measures ............ . 94
15.1 Some extensions to general Borel measures 97
15.2 Regular Borel measures and Radon measures 97
16 Regular outer measures and Radon measures 98
16.1 More on Radon measures .. 99
17 Vitali coverings . . . . . . . . . . 99
18 The Besicovitch covering theorem 103
19 Proof of Proposition 18.2. . . . . 105
20 The Besicovitch measure-theoretical covering theorem 107
Problems and Complements ............... . 110
III The Lebesgue Integral 123
Measurable functions . 123
2 The Egorov theorem . . . . . . . 126
2.1 The Egorov theorem in]ftN . 128
2.2 More on Egorov's theorem . 128
3 Approximating measurable functions by simple functions. 128
4 Convergence in measure . . . . . . . . . . . . . 130
5 Quasi-continuous functions and Lusin's theorem 133
6 Integral of simple functions .......... . 135
7 The Lebesgue integral of nonnegative functions . 136
8 Fatou's lemma and the monotone convergence theorem. 137
9 Basic properties of the Lebesgue integral 139
10 Convergence theorems . . . . . . 141
11 Absolute continuity of the integral 142
12 Product of measures . . . . 142
13 On the structure of (A x B) . . . 144
14 The Fubini-Tonelli theorem . . . 147
14.1 The Tonelli version of the Fubini theorem 148
15 Some applications of the Fubini-Tonelli theorem 148
15.1 Integrals in terms of distribution functions . 148
15.2 Convolution integrals .......... . 149
15.3 The Marcinkiewicz integral ...... . 150
16 Signed measures and the Hahn decomposition . 151
17 The Radon-Nikodym theorem . 154
18 Decomposing measures 157
18.1 The Jordan decomposition 157
18.2 The Lebesgue decomposition 159
18.3 A general version of the Radon-Nikodym theorem 160
Problems and Complements ................ . 160
viii Contents
IV Topics on Measurable Functions of Real Variables 171
1 Functions of bounded variations . . . . . . . . 171
2 Dini derivatives . . . . . . . . . . . . . . . . . 173
3 Differentiating functions of bounded variation . 176
4 Differentiating series of monotone functions 177
5 Absolutely continuous functions 179
6 Density of a measurable set .. 181
7 Derivatives of integrals . . . . . 182
8 Differentiating Radon measures 184
9 Existence and measurability of DIl)J 186
9.1 Proof of Proposition 9.2 .. . 188
10 Representing DIl)J ........ . 189
«
10.1 Representing D 11 )J for )J f1 189
10.2 Representing D 1l)J for )J J.. JL . 191
11 The Lebesgue differentiation theorem 191
11.1 Points of density . . . . . . . . 192
11.2 Lebesgue points of an integrable function 192
12 Regular families . 193
13 Convex functions . . . . . . . . 194
14 Jensen's inequality ...... . 196
15 Extending continuous functions 197
16 The Weierstrass approximation theorem 199
17 The Stone-Weierstrass theorem .... .200
18 Proof of the Stone-Weierstrass theorem · 201
18.1 Proof of Stone's theorem .. .202
19 The Ascoli-Arzela theorem '" · 203
19.1 Precompact subsets of e(l) .204
Problems and Complements · 205
V The LP(E) Spaces 221
Functions in LP (E) and their norms · 221
1.1 The spaces LP for 0 < p < 1 .222
1.2 The spaces L q for q < 0 . . . .222
2 The Holder and Minkowski inequalities 223
3 The reverse Holder and Minkowski inequalities .224
4 More on the spaces LP and their norms . . . . · 225
4.1 Characterizing the norm II f II P for 1 :::: p < 00 · 225
4.2 The norm II . 1100 for E of finite measure . . . . .226
4.3 The continuous version of the Minkowski inequality · 227
5 L P (E) for 1 :::: p :::: 00 as normed spaces of equivalence classes .227
5.1 LP(E) for 1 :::: p :::: 00 as a metric topological vector space. · 228
6 A metric topology for LP(E) when 0 < p < 1 .229
6.1 Open convex subsets of LP(E) when 0 < p < 1 .229
7 Convergence in LP(E) and completeness .230
8 Separating V' (E) by simple functions. . . . . . . . . .232
Contents ix
9 Weak convergence in LP (E) . . . . . . . . . . . . . 234
9.1 A counterexample .............. . 234
10 Weak lower semicontinuity of the norm in LP (E) . . 235
11 Weak convergence and norm convergence . . . . 236
11.1 Proof of Proposition 11.1 for p ::: 2 . . . . 237
11.2 Proof of Proposition 11.1 for 1 < P < 2 . . 237
12 Linear functionals in LP(E) . . . . . . . . . . . 238
13 The Riesz representation theorem . . . . . . . . 239
13.1 Proof of Theorem 13.1: The case where {X, A, ttl is finite . 240
13.2 Proof of Theorem 13.1: The case where {X, A, p,} is a-finite. 241
13.3 Proof of Theorem 13.1: The case where 1 < p < 00 . 242
14 The Hanner and Clarkson inequalities . 243
14.1 Proof of Hanner's inequalities . . . . . . 244
14.2 Proof of Clarkson's inequalities . . . . . 245
15 Uniform convexity of LP(E) for 1 < P < 00 . 246
16 The Riesz representation theorem by uniform convexity . 247
16.1 Proof of Theorem 13.1: The case where 1 < P < 00 . 247
16.2 The case where p = 1 and E is of finite measure . 248
16.3 The case where p = 1 and {X, A, p,} is a-finite. . 249
17 Bounded linear functional in LP (E) for 0 < p < 1 . . 250
17.1 An alternate proof of Proposition 17.1 . . . . . . 250
18 If E c]RN and p E [1, (0), then U(E) is separable . 251
18.1 L''''(E) is not separable ............ . 254
19 Selecting weakly convergent subsequences . . . . . . 254
20 Continuity ofthe translation in LP(E) for 1 :::: p < 00 . 255
21 Approximating functions in LP(E) with functions in COO (E) . . 257
22 Characterizing precompact sets in LP (E) . 260
Problems and Complements ......... . 262
VI Banach Spaces 275
Normed spaces . . . . . . . . . . . . . . . . . .275
1.1 Serninorms and quotients . . . . . . . . . .276
2 Finite-and infinite-dimensional normed spaces .277
2.1 A counterexample .... .277
2.2 The Riesz lemma . . . . . . 278
2.3 Finite-dimensional spaces . 279
3 Linear maps and functionals . . . 280
4 Examples of maps and functionals . 282
4.1 Functionals......... . 283
4.2 Linear functionals on c(E) . 283
5 Kernels of maps and functionals . . 284
6 Equibounded families of linear maps . 285
6.1 Another proof of Proposition 6.1 . 286
7 Contraction mappings ........ . 286
7.1 Applications to some Fredholm integral equations. . 287
x Contents
8 The open mapping theorem · 288
8.1 Some applications .. · 289
8.2 The closed graph theorem · 289
9 The Hahn-Banach theorem. . . .290
10 Some consequences of the Hahn-Banach theorem .292
10.1 Tangent planes . . . . . . · 295
II Separating convex subsets of X ...... . · 295
12 Weak topologies .............. . · 297
12.1 Weakly and strongly closed convex sets · 299
13 Reflexive Banach spaces . . . . . . .300
14 Weak compactness . . . . . . . . . · 301
14.1 Weak sequential compactness · 302
15 The weak* topology · 303
16 The Alaoglu theorem . . . . . .304
17 Hilbert spaces . . . . . . . . . .306
17.1 The Schwarz inequality. .307
17.2 The parallelogram identity .307
18 Orthogonal sets, representations, and functionals · 308
18.1 Bounded linear functionals on H . · 310
19 Orthonormal systems . . . . . · 310
19.1 The Bessel inequality ... · 311
19.2 Separable Hilbert spaces · 312
20 Complete orthonormal systems · 312
20.1 Equivalent notions of complete systems · 313
20.2 Maximal and complete orthonormal systems · 313
20.3 The Gram-Schmidt orthonormalization process · 314
20.4 On the dimension of a separable Hilbert space. · 314
Problems and Complements .............. . · 314
VII Spaces of Continuous Functions, Distributions, and Weak
Derivatives 325
Spaces of continuous functions. . . . . . 325
1.1 Partition of unity . . . . . . . . . . 326
2 Bounded linear functionals on Co(lj{N) . 327
2.1 Remarks on functionals of the type (2.2) and (2.3) . 327
2.2 Characterizing CoOItN)* . . . . . . . . . . . . 328
3 Positive linearfunctionals on CoORN) . . . . . . . . 328
4 Proof of Theorem 3.3: Constructing the measure JJ- • 331
5 Proof of Theorem 3.3: Representing T as in (3.3) . . 333
6 Characterizing bounded linear functionals on CoOItN) . 335
6.1 Locally bounded linear functionals on CoOItN) . 335
6.2 Bounded linear functionals on CoORN) .. . 336
7 A topology for C;:'(E) for an open set E C]RN . . 337
8 A metric topology for C;:' (E) . . . . . 339
8. I Equivalence of these topologies . . . . . . . 340