Real Analysis, Second Edition

REAL ANALYSIS, 2ND ED.......Page 1
Title Page......Page 3
Preface......Page 4
Contents......Page 6
Prologue to the Student......Page 12
1 Introduction......Page 16
2 Functions......Page 19
3 Unions, Intersections, and Complements......Page 22
4 Algebras of Sets......Page 27
5 The Axiom of Choice and Infinite Direct Products......Page 29
6 Countable Sets......Page 30
7 Relations and Equivalences......Page 33
8 Partial Orderings and the Maximal Principle......Page 34
9 Well Ordering and the Countable Ordinals......Page 35
Part I: Theory of Functions of a Real Variable......Page 38
1 Axioms for the Real Numbers......Page 40
2 The Natural and Rational Numbers as Subsets of R......Page 43
3 The Extended Real Numbers......Page 45
4 Sequences of Real Numbers......Page 46
5 Open and Closed Sets of Real Numbers......Page 49
6 Continuous Functions......Page 55
7 Borel Sets......Page 61
1 Introduction......Page 63
2 Outer Measure......Page 65
3 Measurable Sets and Lebesgue Measure......Page 67
*4 A Nonmeasurable Set......Page 74
5 Measurable Functions......Page 76
6 Littlewood's Three Principles......Page 82
1 The Riemann Integral......Page 84
2 The Lebesgue Integral of a Bounded Function over a Set of Finite Measure......Page 86
3 The Integral of a Nonnegative Function......Page 93
4 The General Lebesgue Integral......Page 97
*5 Convergence in Measure......Page 102
1 Differentiation of Monotone Functions......Page 105
2 Functions of Bounded Variation......Page 109
3 Differentiation of an Integral......Page 112
4 Absolute Continuity......Page 115
*5 Convex Functions......Page 119
1 The L[sup(p)] Spaces......Page 122
2 The Hölder and Minkowski Inequalities......Page 123
3 Convergence and Completeness......Page 126
4 Bounded Linear Functionals on the L[sup(p)] Spaces......Page 130
Part II: Abstract Spaces......Page 136
1 Introduction......Page 138
2 Open and Closed Sets......Page 140
3 Continuous Functions and Homeomorphisms......Page 142
4 Convergence and Completeness......Page 144
5 Uniform Continuity and Uniformity......Page 146
6 Subspaces......Page 148
7 Baire Category......Page 150
1 Fundamental Notions......Page 153
2 Bases and Countability......Page 156
3 The Separation Axioms and Continuous Real-Valued Functions......Page 158
4 Product Spaces......Page 161
5 Connectedness......Page 163
*6 Absolute G[sub(δ)]'s......Page 165
*7 Nets......Page 166
1 Basic Properties......Page 168
2 Countable Compactness and the Bolzano–Weierstrass Property......Page 170
3 Compact Metric Spaces......Page 174
4 Products of Compact Spaces......Page 176
5 Locally Compact Spaces......Page 179
*6 The Stone–Čech Compactification......Page 181
7 The Stone–Weierstrass Theorem......Page 182
*8 The Ascoli Theorem......Page 188
1 Introduction......Page 192
2 Linear Operators......Page 195
3 Linear Functionals and the Hahn–Banach Theorem......Page 197
4 The Closed Graph Theorem......Page 204
*5 Topological Vector Spaces......Page 208
*6 Weak Topologies......Page 211
*7 Convexity......Page 214
8 Hilbert Space......Page 221
Part III: General Measure and Integration Theory......Page 226
1 Measure Spaces......Page 228
2 Measurable Functions......Page 234
3 Integration......Page 236
*4 General Convergence Theorems......Page 242
5 Signed Measures......Page 243
6 The Radon–Nikodym Theorem......Page 249
7 The L[sup(p)] Spaces......Page 254
1 Outer Measure and Measurability......Page 261
2 The Extension Theorem......Page 264
*3 The Lebesgue–Stieltjes Integral......Page 272
4 Product Measures......Page 275
*5 Inner Measure......Page 285
*6 Extension by Sets of Measure Zero......Page 292
*7 Carathéodory Outer Measure......Page 294
1 Introduction......Page 297
2 The Extension Theorem......Page 299
3 Uniqueness......Page 305
4 Measurability and Measure......Page 306
1 Baire Sets and Borel Sets......Page 312
2 Positive Linear Functionals and Baire Measures......Page 315
3 Bounded Linear Functionals on C(X)......Page 319
*4 The Borel Extension of a Measure......Page 324
1 Point Mappings and Set Mappings......Page 328
2 Measure Algebras......Page 330
3 Borel Equivalences......Page 335
4 Set Mappings and Point Mappings on Complete Metric Spaces......Page 339
5 The Isometries of L[sup(p)]......Page 342
Epilogue......Page 346
Bibliography......Page 348
Index of Symbols......Page 350
Subject Index......Page 352