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Ⅰ Lebesgue Integration for Functions of a Single Real Variable 1

Preliminaries on Sets,Mappings,and Relations 3

Unions and Intersections of Sets 3

Equivalence Relations,the Axiom of Choice,and Zorn's Lemma 5

1 The Real Numbers:Sets,Sequences,and Functions 7

1.1 The Field,Positivity,and Completeness Axioms 7

1.2 The Natural and Rational Numbers 11

1.3 Countable and Uncountable Sets 13

1.4 Open Sets,Closed Sets,and Borel Sets of Real Numbers 16

1.5 Sequences of Real Numbers 20

1.6 Continuous Real-Valued Functions of a Real Variable 25

2 Lebesgue Measure 29

2.1 Introduction 29

2.2 Lebesgue Outer Measure 31

2.3 The σ-Algebra of Lebesgue Measurable Sets 34

2.4 Outer and Inner Approximation of Lebesgue Measurable Sets 40

2.5 Countable Additivity,Continuity,and the Borel-Cantelli Lemma 43

2.6 Nonmeasurable Sets 47

2.7 The Cantor Set and the Cantor-Lebesgue Function 49

3 Lebesgue Measurable Functions 54

3.1 Sums,Products,and Compositions 54

3.2 Sequential Pointwise Limits and Simple Approximation 60

3.3 Littlewood's Three Principles,Egoroff's Theorem,and Lusin's Theorem 64

4 Lebesgue Integration 68

4.1 The Riemann Integral 68

4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure 71

4.3 The Lebesgue Integral of a Measurable Nonnegative Function 79

4.4 The General Lebesgue Integral 85

4.5 Countable Additivity and Continuity of Integration 90

4.6 Uniform Integrability:The Vitali Convergence Theorem 92

5 Lebesgne Integration:Further Topics 97

5.1 Uniform Integrability and Tightness:A General Vitali Convergence Theorem 97

5.2 Convergence in Measure 99

5.3 Characterizations of Riemann and Lebesgue Integrability 102

6 Differentiation and Integration 107

6.1 Continuity of Monotone Functions 108

6.2 Differentiability of Monotone Functions:Lebesgue's Theorem 109

6.3 Functions of Bounded Variation:Jordan's Theorem 116

6.4 Absolutely Continuous Functions 119

6.5 Integrating Derivatives:Differentiating Indefinite Integrals 124

6.6 Convex Functions 130

7 The LP Spaces:Completeness and Approximation 135

7.1 Normed Linear Spaces 135

7.2 The Inequalities of Young,H？lder,and Minkowski 139

7.3 LP Is Complete:The Riesz-Fischer Theorem 144

7.4 Approximation and Separability 150

8 The LP Spaces:Duality and Weak Convergence 155

8.1 The Riesz Representation for the Dual of LP,1？p＜∞ 155

8.2 Weak Sequential Convergence in LP 162

8.3 Weak Sequential Compactness 171

8.4 The Minimization of Convex Functionals 174

Ⅱ　Abstract Spaces:Metric,Topological,Banach,and Hilbert Spaces 181

9 Metric Spaces:General Properties 183

9.1 Examples of Metric Spaces 183

9.2 Open Sets,Closed Sets,and Convergent Sequences 187

9.3 Continuous Mappings Between Metric Spaces 190

9.4 Complete Metric Spaces 193

9.5 Compact Metric Spaces 197

9.6 Separable Metric Spaces 204

10 Metric Spaces:Three Fundamental Theorems 206

10.1 The Arzelà-Ascoli Theorem 206

10.2 The Baire Category Theorem 211

10.3 The Banach Contraction Principle 215

11 Topological Spaces:General Properties 222

11.1 Open Sets,Closed Sets,Bases,and Subbases 222

11.2 The Separation Properties 227

11.3 Countability and Separability 228

11.4 Continuous Mappings Between Topological Spaces 230

11.5 Compact Topological Spaces 233

11.6 Connected Topological Spaces 237

12 Topological Spaces:Three Fundamental Theorems 239

12.1 Urysohn's Lemma and the Tietze Extension Theorem 239

12.2 The Tychonoff Product Theorem 244

12.3 The Stone-Weierstrass Theorem 247

13 Continuous Linear Operators Between Banach Spaces 253

13.1 Normed Linear Spaces 253

13.2 Linear Operators 256

13.3 Compactness Lost:Infinite Dimensional Normed Linear Spaces 259

13.4 The Open Mapping and Closed Graph Theorems 263

13.5 The Uniform Boundedness Principle 268

14 Duality for Normed Linear Spaces 271

14.1 Linear Functionals,Bounded Linear Functionals,and Weak Topologies 271

14.2 The Hahn-Banach Theorem 277

14.3 Reflexive Banach Spaces and Weak Sequential Convergence 282

14.4 Locally Convex Topological Vector Spaces 286

14.5 The Separation of Convex Sets and Mazur's Theorem 290

14.6 The Krein-Milman Theorem 295

15 Compactness Regained:The Weak Topology 298

15.1 Alaoglu's Extension of Helley's Theorem 298

15.2 Reflexivity and Weak Compactness:Kakutani's Theorem 300

15.3 Compactness and Weak Sequential Compactness:The Eberlein-？mulian Theorem 302

15.4 Metrizability of Weak Topologies 305

16 Continuous Linear Operators on Hilbert Spaces 308

16.1 The Inner Product and Orthogonality 309

16.2 The Dual Space and Weak Sequential Convergence 313

16.3 Bessel's Inequality and Orthonormal Bases 316

16.4 Adjoints and Symmetry for Linear Operators 319

16.5 Compact Operators 324

16.6 The Hilbert-Schmidt Theorem 326

16.7 The Riesz-Schauder Theorem:Characterization of Fredhohn Operators 329

Ⅲ　Measure and Integration:General Theory 335

17 General Measure Spaces:Their Properties and Construction 337

17.1 Measures and Measurable Sets 337

17.2 Signed Measures:The Hahn and Jordan Decompositions 342

17.3 The Carathéodory Measure Induced by an Outer Measure 346

17.4 The Construction of Outer Measures 349

17.5 The Carathéodory-Hahn Theorem:The Extension of a Premeasure to a Measure 352

18 Integration Over General Measure Spaces 359

18.1 Measurable Functions 359

18.2 Integration of Nonnegative Measurable Functions 365

18.3 Integration of General Measurable Functions 372

18.4 The Radon-Nikodym Theorem 381

18.5 The Nikodym Metric Space:The Vitali-Hahn-Saks Theorem 388

19 General LP Spaces:Completeness,Duality,and Weak Convergence 394

19.1 The Completeness of LP(X,μ),1≤p≤∞ 394

19.2 The Riesz Representation Theorem for the Dual of LP(X,μ),1≤p≤∞ 399

19.3 The Kantorovitch Representation Theorem for the Dual of L∞(X,μ) 404

19.4 Weak Sequential Compactness in LP(X,μ),1<p<1 407

19.5 Weak Sequential Compactness in L1(X,μ):The Dunford-Pettis Theorem 409

20 The Construction of Particular Measures 414

20.1 Product Measures:The Theorems of Fubini and Tonelli 414

20.2 Lebesgue Measure on Euclidean Space Rn 424

20.3 Cumulative Distribution Functions and Borel Measures on R 437

20.4 Carathéodory Outer Measures and Hausdorff Measures on a Metric Space 441

21 Measure and Topology 446

21.1 Locally Compact Topological Spaces 447

21.2 Separating Sets and Extending Functions 452

21.3 The Construction of Radon Measures 454

21.4 The Representation of Positive Linear Functionals on Cc(X):The Riesz-Markov Theorem 457

21.5 The Riesz Representation Theorem for the Dual of C(X) 462

21.6 Regularity Properties of Baire Measures 470

22 Invariant Measures 477

22.1 Topological Groups:The General Linear Group 477

22.2 Kakutani's Fixed Point Theorem 480

22.3 Invariant Borel Measures on Compact Groups:yon Neumann's Theorem 485

22.4 Measure Preserving Transformations and Ergodicity:The Bogoliubov-Krilov Theorem 488

Bibliography 495

Index 497