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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