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Real Analysis
 
Contents iii *
 
 
Preface   
xiii
 
 
VOLUME ONE                                    


 
1 BACKGROUND AND PREVIEW
 1
1.1 The Real Numbers 2
1.2 Compact Sets of Real Numbers 7
1.3 Countable Sets 10
1.4 Uncountable Cardinals  12
1.5 Transfinite Ordinals 14
1.6 Category 18
1.7 Outer Measure and Outer Content 20
1.8 Small Sets 22
1.9 Measurable Sets of Real Numbers 25
1.10 Nonmeasurable Sets 29
1.11 Zorn’s Lemma 32
1.12 Borel Sets of Real Numbers 34
1.13 Analytic Sets of Real Numbers 35
1.14 Bounded Variation 37
1.15 Newton’s Integral 40
1.16 Cauchy’s Integral  41
1.17 Riemann’s Integral 43
1.18 Volterra’s Example 45
1.19 Riemann–Stieltjes Integral 47
1.20 Lebesgue’s Integral 50
1.21 The Generalized Riemann Integral 52
1.22 Additional Problems for Chapter 1 55
2  MEASURE SPACES 58
2.1 One-Dimensional Lebesgue Measure 59
2.2 Additive Set Functions 64
2.3 Measures and Signed Measures 69
2.4 Limit Theorems 73
2.5 The Jordan and Hahn Decomposition Theorems 76
2.6 Hahn Decomposition 78
2.7 Complete Measures 80
2.8 Outer Measures 82
2.9 Method I 87
2.10 Regular Outer Measures 89
2.11 Nonmeasurable Sets 93
2.12 More About Method I 96
2.13 Completions 99
2.14 Additional Problems for Chapter 2 102
3  METRIC OUTER MEASURES
 105
3.1 Metric Space 105
3.2 Measures on Metric Spaces 110
3.3 Method II 115
3.4 Approximations 118
3.5 Construction of Lebesgue–Stieltjes Measures 121
3.6 Properties of Lebesgue–Stieltjes Measures 126
3.7 Lebesgue–Stieltjes Measures in Rn 131
3.8 Hausdorff Measures and Hausdorff Dimension 133
3.9 Methods III and IV 142
3.10 Mini-Vitali Theorem 145
3.11 Lebesgue differentiation theorem 149
3.12 Additional Remarks on Special Sets  154
3.13 Additional Problems for Chapter 3 158
4  MEASURABLE FUNCTIONS
 162
4.1 Definitions and Basic Properties 163
4.2 Sequences of Measurable Functions  168
4.3 Egoroff’s Theorem 173
4.4 Approximations by Simple Functions 176
4.5 Approximation by Continuous Functions 180
4.6 Additional Problems for Chapter 4 184
5  INTEGRATION 188
5.1 Introduction
5.2 Integrals of Nonnegative Functions 193
5.3 Fatou’s Lemma 197
5.4 Integrable Functions 201
5.5 Riemann and Lebesgue 204
5.6 Countable Additivity of the Integral 212
5.7 Absolute Continuity 215
5.8 Radon–Nikodým Theorem 220
5.9 Convergence Theorems 227
5.10 Relations to Other Integrals  232
5.11 Integration of Complex Functions 237
5.12 Additional Problems for Chapter 5 240
6  FUBINI'S THEOREM
 245
6.1 Product Measures 246
6.2 Fubini’s Theorem 254
6.3 Tonelli’s Theorem 255
6.4 Additional Problems for Chapter 6 257
7  DIFFERENTIATION 259
7.1 The Vitali Covering Theorem 259
7.2 Lebesgue’s Differentiation Theorem 267
7.3 The Banach–Zarecki Theorem 271
7.4 Determining a Function by a Derivative 274
7.5 Calculating a Function from a Derivative
 276
7.6 Total Variation of a Function 282
7.7 Approximate Continuity and Lebesgue Points 292
7.8 Additional Problems for Chapter 7 297
8  DIFFERENTIATION OF MEASURES 304
8.1 Differentiation of Lebesgue–Stieltjes Measures  304
8.2 The Cube Basis  309
8.3 Lebesgue Decomposition Theorem  314
8.4 The Interval Basis  316
8.5 Net Structures  323
8.6 Radon–Nikodým Derivative in a Measure Space 328
8.7 Summary, Comments, and References  333
8.8 Additional Problems for Chapter 8  339
INDEX * 631





VOLUME TWO


9  METRIC SPACES 341
9.1 Definitions and Examples  341
9.2 Convergence and Related Notions  340
9.3 Continuity  354
9.4 Homeomorphisms and Isometries  356
9.5 Separable Spaces  359
9.6 Complete Spaces  361
9.7 Contraction Maps  366
9.8 Applications  368
9.9 Compactness  376
9.10 Totally Bounded Spaces  377
9.11 Compact Sets in C(X)  378
9.12 Application of the Arzelà–Ascoli Theorem  382
9.13 The Stone–Weierstrass Theorem  384
9.14 The Isoperimetric Problem  387
9.15 More on Convergence  390
9.16 Additional Problems for Chapter 9  393
10  BAIRE CATEGORY  396
10.1 The Banach-Mazur Game on the Real Line  396
10.2 The Baire Category Theorem 398
10.3 The Banach–Mazur Game  401
10.4 The First Classes of Baire and Borel  406
10.5 Properties of Baire-1 Functions  411
10.6 Topologically Complete Spaces  415
10.7 Applications to Function Spaces  419
10.8 Additional Problems for Chapter 10  429
11 ANALYTIC SETS 434
11.1 Products of Metric Spaces  434
11.2 Baire Space  436
11.3 Analytic Sets  439
11.4 Borel Sets  442
11.5 An Analytic Set That Is Not Borel  446
11.6 Measurability of Analytic Sets  448
11.7 The Suslin Operation  450
11.8 A Method to Show a Set Is Not Borel
452
11.9 Differentiable Functions  455
11.10 Additional Problems for Chapter 11
458
 
 
12 BANACH SPACES
461
12.1 Normed Linear Spaces
461
12.2 Compactness
466
12.3 Linear Operators
470
12.4 Banach Algebras
474
12.5 The Hahn–Banach Theorem
477
12.6 Improving Lebesgue Measure
481
12.7 The Dual Space
486
12.8 The Riesz Representation Theorem
489
12.9 Separation of Convex Sets
494
12.10 An Embedding Theorem
499
12.11 Uniform Boundedness Principle
501
12.12 An Application to Summability
504
12.13 The Open Mapping Theorem
508
12.14 The Closed Graph Theorem
511
12.15 Additional Problems for Chapter 12
514
 
 
13 THE LP  SPACES
516
13.1 The Basic Inequalities
516
13.2 The ℓp and Lp Spaces (1 ≤ p < ∞)
520
13.3 The Spaces ℓ∞ and L∞
522
13.4 Separability
524
13.5 The Spaces ℓ2 and L2
527
13.6 Continuous Linear Functionals on Lp(μ)
532
13.7 The Lp Spaces (0 < p < 1)
535
13.8 Relations
538
13.9 The Banach Algebra L1(R)
540
13.10 Weak Sequential Convergence
545
13.11 Closed Subspaces of the Lp Spaces
548
13.12 Additional Problems for Chapter 13
551
 
 
14 HILBERT SPACES
553
14.1 Inner Products
554
14.2 Convex Sets
559
14.3 Continuous Linear Functionals
561
14.4 Orthogonal Series
563
14.5 Weak Sequential Convergence
569
14.6 Compact Operators
672
14.7 Projections
576
14.8 Eigenvectors and Eigenvalues
578
14.9 Spectral Decomposition
583
14.10 Additional Problems for Chapter 14
586
 
 
15 FOURIER SERIES
589
15.1 Notation and Terminology
590
15.2 Dirichlet’s Kernel
594
15.3 Fejér’s Kernel
597
15.4 Convergence of the Cesàro Means
600
15.5 The Fourier Coefficients
604
15.6 Weierstrass Approximation Theorem
607
15.7 Pointwise Convergence
609
15.8 Pointwise Convergence: Dini’s Test
614
15.9 Pointwise Divergence
616
15.10 Characterizations
618
15.11 Fourier Series in Hilbert Space
620
15.12 Riemann’s Theorems
622
15.13 Cantor’s Uniqueness Theorem
625
 15.14 Additional Problems for Chapter 15
629
 
 
Index *
631
 
 * The complete Table of Contents and Index are included in each volume.

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