Real analysis

Title page......Page 1
Date-line......Page 2
Preface to the Third Edition......Page 3
Preface to the Second Edition......Page 7
Contents......Page 9
Prologue to the Student......Page 14
1 Introduction......Page 19
2 Functions......Page 22
3 Unions, intersections, and complements......Page 25
4 Algebras of sets......Page 30
5 The axiom of choice and infinite direct products......Page 32
6 Countable sets......Page 33
7 Relations and equivalences......Page 36
8 Partial orderings and the maximal principle......Page 37
9 Well ordering and the countable ordinals......Page 39
Part One. THEORY OF FUNCTIONS OF A REAL VARIABLE......Page 42
1 Axioms for the real numbers......Page 44
2 The natural and rational numbers as subsets of $\\mathbb{R}$......Page 47
3 The extended real numbers......Page 49
4 Sequences of real numbers......Page 50
5 Open and closed sets of real numbers......Page 53
6 Continuous functions......Page 60
7 Borel sets......Page 65
1 Introduction......Page 67
2 Outer measure......Page 69
3 Measurable sets and Lebesgue measure......Page 71
*4 A nonmeasurable set......Page 77
5 Measurable functions......Page 79
6 Littlewood's three principles......Page 85
1 The Riemann integral......Page 88
2 The Lebesgue integral of a bounded function over a set of finite measure......Page 90
3 The integral of a nonnegative function......Page 98
4 The general Lebesgue integral......Page 102
*5 Convergence in measure......Page 108
1 Differentiation of monotone functions......Page 110
2 Functions of bounded variation......Page 115
3 Differentiation of an integral......Page 117
4 Absolute continuity......Page 121
5 Convex functions......Page 126
1 The $L^p$ spaces......Page 131
2 The Minkowski and Holder inequalities......Page 132
3 Convergence and completeness......Page 136
4 Approximation in $L^p$......Page 140
5 Bounded linear functionals on the $L^p$ spaces......Page 143
Part Two. ABSTRACT SPACES......Page 150
1 Introduction......Page 152
2 Open and closed sets......Page 154
3 Continuous functions and homeomorphisms......Page 157
4 Convergence and completeness......Page 159
5 Uniform continuity and uniformity......Page 161
6 Subspaces......Page 164
7 Compact metric spaces......Page 165
8 Baire category......Page 171
9 Absolute $G_\\delta$'s......Page 177
10 The Ascoli-Arzela Theorem......Page 180
1 Fundamental notions......Page 184
2 Bases and countability......Page 188
3 The separation axioms and continuous real-valued functions......Page 191
4 Connectedness......Page 195
5 Products and direct unions of topological spaces......Page 197
*6 Topological and uniform properties......Page 200
*7 Nets......Page 201
1 Compact spaces......Page 203
2 Countable compactness and the Bolzano-Weierstrass property......Page 206
3 Products of compact spaces......Page 209
4 Locally compact spaces......Page 212
5 $\\sigma$-compact spaces......Page 216
*6 Paracompact spaces......Page 217
7 Manifolds......Page 219
*8 The Stone-Cech compactification......Page 222
9 The Stone-Weierstrass Theorem......Page 223
1 Introduction......Page 230
2 Linear operators......Page 233
3 Linear functional and the Hahn-Banach Theorem......Page 235
4 The Closed Graph Theorem......Page 237
5 Topological vector spaces......Page 246
6 Weak topologies......Page 249
7 Convexity......Page 252
8 Hilbert space......Page 258
Part Three. GENERAL MEASURE AND INTEGRATION THEORY......Page 264
1 Measure spaces......Page 266
2 Measurable functions......Page 272
3 Integration......Page 276
4 General Convergence Theorems......Page 281
5 Signed measures......Page 283
6 The Radon-Nikodym Theorem......Page 289
7 The $L^p$-spaces......Page 295
1 Outer measure and measurability......Page 301
2 The Extension Theorem......Page 304
3 The Lebesgue-Stieltjes integral......Page 312
4 Product measures......Page 316
5 Integral operators......Page 326
*6 Inner measure......Page 330
*7 Extension by sets of measure zero......Page 338
8 Caratheodory outer measure......Page 339
9 HausdorfT measure......Page 342
1 Baire sets and Borel sets......Page 344
2 The regularity of Baire and Borel measures......Page 350
3 The construction of Borel measures......Page 358
4 Positive linear functional and Borel measures......Page 365
5 Bounded linear functional on $C(X)$......Page 368
1 Homogeneous spaces......Page 374
2 Topological equicontinuity......Page 375
3 The existence of invariant measures......Page 378
4 Topological groups......Page 383
5 Group actions and quotient spaces......Page 389
6 Unicity of invariant measures......Page 391
7 Groups of diffeomorphisms......Page 401
1 Point mappings and set mappings......Page 405
2 Boolean $\\sigma$-algebras......Page 407
3 Measure algebras......Page 411
4 Borel equivalences......Page 414
5 Borel measures on complete separable metric spaces......Page 419
6 Set mappings and point mappings on complete separable metric spaces......Page 425
7 The isometries of $L^p$......Page 428
1 Introduction......Page 432
2 The Extension Theorem......Page 435
3 Uniqueness......Page 440
4 Measurability and measure......Page 442
Bibliography......Page 448
Index of Symbols......Page 450
Subject Index......Page 452