1 Set Theory 1
1 Introduction 1
2 Functions 3
3 Unions，intersections，and complements 6
4 Algebras of sets 11
5 The axiom of choice and infinite direct products 13
6 Countable sets 13
7 Relations and equivalences 16
8 Partial orderings and the maximal principle 18
Part One THEORY OF FUNCTIONS OF A REAL VARIABLE 19
2 The Real Number System 21
1 Axioms for the real numbers 21
2 The natural and rational numbers as subsets of R 24
3 The extended real numbers 26
4 Sequences of real numbers 26
5 Open and closed sets of real numbers 30
6 Continuous functions 36
7 Borel sets 41
3 Lebesgue Measure 43
1 Introduction 43
2 Outer measure 44
3 Measurable sets and Lebesgue measure 47
4 A nonmeasurable set 52
5 Measurable functions 54
6 Littlewood＇s three principles 59
4 The Lebesgue Integral 61
1 The Riemann integral 61
2 The Lebesgue integral of a bounded function over a set of finite measure 63
3 The integral of a nonnegative function 70
4 The general Lebesgue integral 75
5 Convergence in measure 78
5 Differentiation and Integration 80
1 Differentiation of monotone functions 80
2 Functions of bounded variation 84
3 Differentiation of an integral 86
4 Absolute continuity 90
6 The Classical Banach Spaces 93
1 The Lv spaces 93
2 The Holder and Minkowski inequalities 94
3 Convergence and completeness 97
4 Bounded linear functionals on the Lv spaces 101
Epilogue to Part One 106
Part Two ABSTRACT SPACES 107
7 Metric Spaces 109
1 Introduction 109
2 Open and closed sets 111
3 Continuous functions and homeomorphisms 113
4 Convergence and completeness 115
5 Uniform continuity and uniformity 117
6 Subspaces 119
7 Baire category 121
8 Topological Spaces 124
1 Fundamental notions 124
2 Bases and countability 127
3 The separation axioms and continuous real-valued functions 130
4 Connectedness 133
5 Nets 134
9 Compact Spaces 136
1 Basic properties 136
2 Countable compactness and the Bolzano-Weierstrass property 138
3 Compact metric spaces 141
4 Products of compact spaces 143
5 Locally compact spaces 146
6 The Stone-Weierstrass theorem 147
7 The Ascoli theorem 153
10 Banach Spaces 157
1 Introduction 157
2 Linear operators 160
3 Linear functionals and the Hahn-Banach theorem 162
4 The closed graph theorem 169
5 Weak topologies 172
6 Convexity 175
7 Hilbert space 183
Epilogue to Part Two 188
Part Three GENERAL MEASURE AND INTEGRATION THEORY 189
11 Measure and Integration 191
1 Measure spaces 191
2 Measurable functions 195
3 Integration 196
4 Signed measures 202
5 The Radon-Nikodym theorem 207
12 Measure and Outer Measure 216
1 Outer measure and measurability 216
2 The extension theorem 219
3 The Lebesgue-Stieltjes integral 225
4 Product measures 229
5 Carathéodory outer measure 235
13 The Daniell Integral 238
1 Introduction 238
2 The extension theorem 239
3 Measurability 244
4 Uniqueness 247
5 Measure and topology 250
6 Bounded linear functionals on C(X) 254
14 Mappings of Measure Spaces 260
1 Point mappings and set mappings 260
2 Measure algebras 262
3 Borel equivalences 267
4 Set mappings and point mappings on the unit interval 271
5 The isometries of Lv 273
Epilogue to Part Three 278
Bibliography 279
Index of Symbols 280
Subject Index 282