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Ⅰ Lebesgue Integration for Functions of a Single Real Variable1

Preliminaries on Sets,Mappings,and Relations3

Unions and Intersections of Sets3

Equivalence Relations,the Axiom of Choice,and Zorn's Lemma5

1 The Real Numbers:Sets,Sequences,and Functions7

1.1 The Field,Positivity,and Completeness Axioms7

1.2 The Natural and Rational Numbers11

1.3 Countable and Uncountable Sets13

1.4 Open Sets,Closed Sets,and Borel Sets of Real Numbers16

1.5 Sequences of Real Numbers20

1.6 Continuous Real-Valued Functions of a Real Variable25

2 Lebesgue Measure29

2.1 Introduction29

2.2 Lebesgue Outer Measure31

2.3 The σ-Algebra of Lebesgue Measurable Sets34

2.4 Outer and Inner Approximation of Lebesgue Measurable Sets40

2.5 Countable Additivity,Continuity,and the Borel-Cantelli Lemma43

2.6 Nonmeasurable Sets47

2.7 The Cantor Set and the Cantor-Lebesgue Function49

3 Lebesgue Measurable Functions54

3.1 Sums,Products,and Compositions54

3.2 Sequential Pointwise Limits and Simple Approximation60

3.3 Littlewood's Three Principles,Egoroff's Theorem,and Lusin's Theorem64

4 Lebesgue Integration68

4.1 The Riemann Integral68

4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure71

4.3 The Lebesgue Integral of a Measurable Nonnegative Function79

4.4 The General Lebesgue Integral85

4.5 Countable Additivity and Continuity of Integration90

4.6 Uniform Integrability:The Vitali Convergence Theorem92

5 Lebesgne Integration:Further Topics97

5.1 Uniform Integrability and Tightness:A General Vitali Convergence Theorem97

5.2 Convergence in Measure99

5.3 Characterizations of Riemann and Lebesgue Integrability102

6 Differentiation and Integration107

6.1 Continuity of Monotone Functions108

6.2 Differentiability of Monotone Functions:Lebesgue's Theorem109

6.3 Functions of Bounded Variation:Jordan's Theorem116

6.4 Absolutely Continuous Functions119

6.5 Integrating Derivatives:Differentiating Indefinite Integrals124

6.6 Convex Functions130

7 The LP Spaces:Completeness and Approximation135

7.1 Normed Linear Spaces135

7.2 The Inequalities of Young,H？lder,and Minkowski139

7.3 LP Is Complete:The Riesz-Fischer Theorem144

7.4 Approximation and Separability150

8 The LP Spaces:Duality and Weak Convergence155

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

8.2 Weak Sequential Convergence in LP162

8.3 Weak Sequential Compactness171

8.4 The Minimization of Convex Functionals174

Ⅱ　Abstract Spaces:Metric,Topological,Banach,and Hilbert Spaces181

9 Metric Spaces:General Properties183

9.1 Examples of Metric Spaces183

9.2 Open Sets,Closed Sets,and Convergent Sequences187

9.3 Continuous Mappings Between Metric Spaces190

9.4 Complete Metric Spaces193

9.5 Compact Metric Spaces197

9.6 Separable Metric Spaces204

10 Metric Spaces:Three Fundamental Theorems206

10.1 The Arzelà-Ascoli Theorem206

10.2 The Baire Category Theorem211

10.3 The Banach Contraction Principle215

11 Topological Spaces:General Properties222

11.1 Open Sets,Closed Sets,Bases,and Subbases222

11.2 The Separation Properties227

11.3 Countability and Separability228

11.4 Continuous Mappings Between Topological Spaces230

11.5 Compact Topological Spaces233

11.6 Connected Topological Spaces237

12 Topological Spaces:Three Fundamental Theorems239

12.1 Urysohn's Lemma and the Tietze Extension Theorem239

12.2 The Tychonoff Product Theorem244

12.3 The Stone-Weierstrass Theorem247

13 Continuous Linear Operators Between Banach Spaces253

13.1 Normed Linear Spaces253

13.2 Linear Operators256

13.3 Compactness Lost:Infinite Dimensional Normed Linear Spaces259

13.4 The Open Mapping and Closed Graph Theorems263

13.5 The Uniform Boundedness Principle268

14 Duality for Normed Linear Spaces271

14.1 Linear Functionals,Bounded Linear Functionals,and Weak Topologies271

14.2 The Hahn-Banach Theorem277

14.3 Reflexive Banach Spaces and Weak Sequential Convergence282

14.4 Locally Convex Topological Vector Spaces286

14.5 The Separation of Convex Sets and Mazur's Theorem290

14.6 The Krein-Milman Theorem295

15 Compactness Regained:The Weak Topology298

15.1 Alaoglu's Extension of Helley's Theorem298

15.2 Reflexivity and Weak Compactness:Kakutani's Theorem300

15.3 Compactness and Weak Sequential Compactness:The Eberlein-？mulian Theorem302

15.4 Metrizability of Weak Topologies305

16 Continuous Linear Operators on Hilbert Spaces308

16.1 The Inner Product and Orthogonality309

16.2 The Dual Space and Weak Sequential Convergence313

16.3 Bessel's Inequality and Orthonormal Bases316

16.4 Adjoints and Symmetry for Linear Operators319

16.5 Compact Operators324

16.6 The Hilbert-Schmidt Theorem326

16.7 The Riesz-Schauder Theorem:Characterization of Fredhohn Operators329

Ⅲ　Measure and Integration:General Theory335

17 General Measure Spaces:Their Properties and Construction337

17.1 Measures and Measurable Sets337

17.2 Signed Measures:The Hahn and Jordan Decompositions342

17.3 The Carathéodory Measure Induced by an Outer Measure346

17.4 The Construction of Outer Measures349

17.5 The Carathéodory-Hahn Theorem:The Extension of a Premeasure to a Measure352

18 Integration Over General Measure Spaces359

18.1 Measurable Functions359

18.2 Integration of Nonnegative Measurable Functions365

18.3 Integration of General Measurable Functions372

18.4 The Radon-Nikodym Theorem381

18.5 The Nikodym Metric Space:The Vitali-Hahn-Saks Theorem388

19 General LP Spaces:Completeness,Duality,and Weak Convergence394

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<1407

19.5 Weak Sequential Compactness in L1(X,μ):The Dunford-Pettis Theorem409

20 The Construction of Particular Measures414

20.1 Product Measures:The Theorems of Fubini and Tonelli414

20.2 Lebesgue Measure on Euclidean Space Rn424

20.3 Cumulative Distribution Functions and Borel Measures on R437

20.4 Carathéodory Outer Measures and Hausdorff Measures on a Metric Space441

21 Measure and Topology446

21.1 Locally Compact Topological Spaces447

21.2 Separating Sets and Extending Functions452

21.3 The Construction of Radon Measures454

21.4 The Representation of Positive Linear Functionals on Cc(X):The Riesz-Markov Theorem457

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

21.6 Regularity Properties of Baire Measures470

22 Invariant Measures477

22.1 Topological Groups:The General Linear Group477

22.2 Kakutani's Fixed Point Theorem480

22.3 Invariant Borel Measures on Compact Groups:yon Neumann's Theorem485

22.4 Measure Preserving Transformations and Ergodicity:The Bogoliubov-Krilov Theorem488

Bibliography495

Index497