Prologue to the Student 1
I Set Theory 6
1 Introduction 6
2 Functions 9
3 Unions, intersections, and complements 12
4 Algebras of sets 17
5 The axiom of choice and infinite direct products 19
6 Countable sets 20
7 Relations and equivalences 23
8 Partial orderings and the maximal principle 24
9 Well ordering and the countable ordinals 26
Part One
THEORY OF FUNCTIONS OF A
REAL VARIABLE
2 The Real Number System 31
1 Axioms for the real numbers 31
2 The natural and rational numbers as subsets of R 34
3 The extended real numbers 36
4 Sequences of real numbers 37
5 Open and closed sets of real numbers 40
6 Continuous functions 47
7 Borel sets 52
3 Lebesgue Measure 54
I Introduction 54
2 Outer measure 56
3 Measurable sets and Lebesgue measure 58
*4 A nonmeasurable set 64
5 Measurable functions 66
6 Littlewood's three principles 72
4 The Lebesgue Integral 75
1 The Riemann integral 75
2 The Lebesgue integral of a bounded function over a set of finite
measure 77
3 The integral of a nonnegative function 85
4 The general Lebesgue integral 89
*5 Convergence in measure 95
S Differentiation and Integration 97
1 Differentiation of monotone functions 97
2 Functions of bounded variation 102
3 Differentiation of an integral 104
4 Absolute continuity 108
5 Convex functions 113
6 The Classical Banach Spaces 118
1 The Lp spaces 118
2 The Minkowski and Holder inequalities 119
3 Convergence and completeness 123
4 Approximation in Lp 127
5 Bounded linear functionals on the Lp spaces 130
Part Two
ABSTRACT SPACES
7 Metric Spaces 139
1 Introduction 139
2 Open and closed sets 141
3 Continuous functions and homeomorphisms 144
4 Convergence and completeness 146
5 Uniform continuity and uniformity 148
6 Subspaces 151
7 Compact metric spaces 152
8 Baire category 158
9 Absolute Gs 164
10 The Ascoli-Arzela Theorem 167
8 Topological Spaces ltl
I Fundamental notions 171
2 Bases and countability 175
3 The separation axioms and continuous real-valued
functions 178
4 Connectedness 182
5 Products and direct unions of topological spaces 184
*6 Topological and uniform properties 187
*7 Nets 188
9 Compact and Locally Compact Spaces 190
I Compact spaces 190
2 Countable compactness and the Bolzano-Weierstrass
property 193
3 Products of compact spaces 196
4 Locally compact spaces 199
5 a-compact spaces 203
*6 Paracompact spaces 204
7 Manifolds 206
*8 The Stone-Cech compactification 209
9 The Stone-Weierstrass Theorem 210
10 Banach Spaces 217
I Introduction 217
2 Linear operators 220
3 Linear functionals and the Hahn-Banach Theorem 222
4 The Closed Graph Theorem 224
5 Topological vector spaces 233
6 Weak topologies 236
7 Convexity 239
8 Hilbert space 245
Part Three
GENERAL MEASURE AND INTEGRATION
THEORY
11 Measure and Integration 253
1 Measure spaces 253
2 Measurable functions 259
3 Integration 263
4 General Convergence Theorems 268
5 Signed measures 270
6 The Radon-Nikodym Theorem 276
7 The Lp-spaces 282
12 Measure and Outer Measure 288
1 Outer measure and measurability 288
2 The Extension Theorem 291
3 The Lebesgue-Stieltjes integral 299
4 Product measures 303
5 Integral operators 313
*6 Inner measure 317
*7 Extension by sets of measure zero 325
8 Caratheodory outer measure 326
9 Hausdorff measure 329
13 Measure and Topology 331
1 Baire sets and Borel sets 331
2 The regularity of Baire and Borel measures 337
3 The construction of Borel measures 345
4 Positive linear functionals and Borel measures 352
5 Bounded linear functionals on C(X) 355
14 Invariant Measures 361
1 Homogeneous spaces 361
2 Topological equicontinuity 362
3 The existence ofinvariant measures 365
4 Topological groups 370
5 Group actions and quotient spaces 376
6 Unicity ofinvariant measures 378
7 Groups ofdiffeomorphisms 388
15 Mappings of Measure Spaces 392
1 Point mappings and set mappings 392
2 Boolean algebras 394
3 Measure algebras 398
4 Borel equivalences 401
5 Borel measures on complete separable metric spaces 406
6 Set mappings and point mappings on complete separable
metric spaces 412
7 The isometries of Lp 415
16 The Daniell Integral 419
1 Introduction 419
2 The Extension Theorem 422
3 Uniqueness 427
4 Measurability and measure 429
Bibliography 435
Index of Symbols 437
Subject Index 439
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