实分析

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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