《Real Analysis》

1Set Theory1

1 Introduction1

2 Functions3

3 Unions，intersections，and complements6

4 Algebras of sets11

5 The axiom of choice and infinite direct products13

6 Countable sets13

7 Relations and equivalences16

8 Partial orderings and the maximal principle18

Part OneTHEORY OF FUNCTIONS OF A REAL VARIABLE19

2The Real Number System21

1 Axioms for the real numbers21

2 The natural and rational numbers as subsets of R24

3 The extended real numbers26

4 Sequences of real numbers26

5 Open and closed sets of real numbers30

6 Continuous functions36

7 Borel sets41

3Lebesgue Measure43

1 Introduction43

2 Outer measure44

3 Measurable sets and Lebesgue measure47

4 A nonmeasurable set52

5 Measurable functions54

6 Littlewood＇s three principles59

4The Lebesgue Integral61

1 The Riemann integral61

2 The Lebesgue integral of a bounded function over a set of finite measure63

3 The integral of a nonnegative function70

4 The general Lebesgue integral75

5 Convergence in measure78

5Differentiation and Integration80

1 Differentiation of monotone functions80

2 Functions of bounded variation84

3 Differentiation of an integral86

4 Absolute continuity90

6The Classical Banach Spaces93

1 The Lv spaces93

2 The Holder and Minkowski inequalities94

3 Convergence and completeness97

4 Bounded linear functionals on the Lv spaces101

Epilogue to Part One106

Part TwoABSTRACT SPACES107

7Metric Spaces109

1 Introduction109

2 Open and closed sets111

3 Continuous functions and homeomorphisms113

4 Convergence and completeness115

5 Uniform continuity and uniformity117

6 Subspaces119

7 Baire category121

8Topological Spaces124

1 Fundamental notions124

2 Bases and countability127

3 The separation axioms and continuous real-valued functions130

4 Connectedness133

5 Nets134

9Compact Spaces136

1 Basic properties136

2 Countable compactness and the Bolzano-Weierstrass property138

3 Compact metric spaces141

4 Products of compact spaces143

5 Locally compact spaces146

6 The Stone-Weierstrass theorem147

7 The Ascoli theorem153

10Banach Spaces157

1 Introduction157

2 Linear operators160

3 Linear functionals and the Hahn-Banach theorem162

4 The closed graph theorem169

5 Weak topologies172

6 Convexity175

7 Hilbert space183

Epilogue to Part Two188

Part ThreeGENERAL MEASURE AND INTEGRATION THEORY189

11Measure and Integration191

1 Measure spaces191

2 Measurable functions195

3 Integration196

4 Signed measures202

5 The Radon-Nikodym theorem207

12Measure and Outer Measure216

1 Outer measure and measurability216

2 The extension theorem219

3 The Lebesgue-Stieltjes integral225

4 Product measures229

5 Carathéodory outer measure235

13The Daniell Integral238

1 Introduction238

2 The extension theorem239

3 Measurability244

4 Uniqueness247

5 Measure and topology250

6 Bounded linear functionals on C(X)254

14Mappings of Measure Spaces260

1 Point mappings and set mappings260

2 Measure algebras262

3 Borel equivalences267

4 Set mappings and point mappings on the unit interval271

5 The isometries of Lv273

Epilogue to Part Three278

Bibliography279

Index of Symbols280

Subject Index282