Real Analysis, Fourth Edition

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface......Page 4
Contents......Page 8
I Lebesgue Integration for Functions of a Single Real Variable......Page 12
Unions and Intersections of Sets......Page 14
Equivalence Relations, the Axiom of Choice, and Zorn's Lemma......Page 16
1.1 The Field, Positivity, and Completeness Axioms......Page 18
1.2 The Natural and Rational Numbers......Page 22
1.3 Countable and Uncountable Sets......Page 24
1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers......Page 27
1.5 Sequences of Real Numbers......Page 31
1.6 Continuous Real-Valued Functions of a Real Variable......Page 36
2.1 Introduction......Page 40
2.2 Lebesgue Outer Measure......Page 42
2.3 The Q-Algebra of Lebesgue Measurable Sets......Page 45
2.4 Outer and Inner Approximation of Lebesgue Measurable Sets......Page 51
2.5 Countable Additivity, Continuity, and the Borel-Cantelli Lemma......Page 54
2.6 Nonmeasurable Sets......Page 58
2.7 The Cantor Set and the Cantor-Lebesgue Function......Page 60
3.1 Sums, Products, and Compositions......Page 65
3.2 Sequential Pointwise Limits and Simple Approximation......Page 71
3.3 Littlewood's Three Principles, Egoroffs Theorem, and Lusin's Theorem......Page 75
4.1 The Riemann Integral......Page 79
4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure......Page 82
4.3 The Lebesgue Integral of a Measurable Nonnegative Function......Page 90
4.4 The General Lebesgue Integral......Page 96
4.5 Countable Additivity and Continuity of Integration......Page 101
4.6 Uniform Integrability: The Vitali Convergence Theorem......Page 103
5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem......Page 108
5.2 Convergence in Measure......Page 110
5.3 Characterizations of Riemann and Lebesgue Integrability......Page 113
6 Differentiation and Integration......Page 118
6.1 Continuity of Monotone Functions......Page 119
6.2 Differentiability of Monotone Functions: Lebesgue's Theorem......Page 120
6.3 Functions of Bounded Variation: Jordan's Theorem......Page 127
6.4 Absolutely Continuous Functions......Page 130
6.5 Integrating Derivatives: Differentiating Indefinite Integrals......Page 135
6.6 Convex Functions......Page 141
7.1 Normed Linear Spaces......Page 146
7.2 The Inequalities of Young, Holder, and Minkowski......Page 150
7.3 1/ Is Complete: The Riesz-Fischer Theorem......Page 155
7.4 Approximation and Separability......Page 161
8.1 The Riesz Representation for the Dual of L^P,1 < p < \infty......Page 166
8.2 Weak Sequential Convergence in L^P......Page 173
8.3 Weak Sequential Compactness......Page 182
8.4 The Minimization of Convex Functionals......Page 185
II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces......Page 192
9.1 Examples of Metric Spaces......Page 194
9.2 Open Sets, Closed Sets, and Convergent Sequences......Page 198
9.3 Continuous Mappings Between Metric Spaces......Page 201
9.4 Complete Metric Spaces......Page 204
9.5 Compact Metric Spaces......Page 208
9.6 Separable Metric Spaces......Page 215
10.1 The Arzelii-Ascoli Theorem......Page 217
10.2 The Baire Category Theorem......Page 222
10.3 The Banach Contraction Principle......Page 226
11.1 Open Sets, Closed Sets, Bases, and Subbases......Page 233
11.2 The Separation Properties......Page 238
11.3 Countability and Separability......Page 239
11.4 Continuous Mappings Between Topological Spaces......Page 241
11.5 Compact Topological Spaces......Page 244
11.6 Connected Topological Spaces......Page 248
12.1 Urysohn's Lemma and the Tietze Extension Theorem......Page 250
12.2 The Tychonoff Product Theorem......Page 255
12.3 The Stone-Weierstrass Theorem......Page 258
13.1 Normed Linear Spaces......Page 264
13.2 Linear Operators......Page 267
13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces......Page 270
13.4 The Open Mapping and Closed Graph Theorems......Page 274
13.5 The Uniform Boundedness Principle......Page 279
14.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies......Page 282
14.2 The Hahn-Banach Theorem......Page 288
14.3 Reflexive Banach Spaces and Weak Sequential Convergence......Page 293
14.4 Locally Convex Topological Vector Spaces......Page 297
14.5 The Separation of Convex Sets and Mazur's Theorem......Page 301
14.6 The Krein-Milman Theorem......Page 306
15.1 Alaoglu's Extension of Helley's Theorem......Page 309
15.2 Reflexivity and Weak Compactness: Kakutani's Theorem......Page 311
15.3 Compactness and Weak Sequential Compactness: The Theorem......Page 313
15.4 Metrizability of Weak Topologies......Page 316
16 Continuous Linear Operators on Hilbert Spaces......Page 319
16.1 The Inner Product and Orthogonality......Page 320
16.2 The Dual Space and Weak Sequential Convergence......Page 324
16.3 Bessel's Inequality and Orthonormal Bases......Page 327
16.4 Adjoints and Symmetry for Linear Operators......Page 330
16.5 Compact Operators......Page 335
16.6 The Hilbert-Schmidt Theorem......Page 337
16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators......Page 340
III Measure and Integration: General Theory......Page 346
17.1 Measures and Measurable Sets......Page 348
17.2 Signed Measures: The Hahn and Jordan Decompositions......Page 353
17.3 The Carath6odory Measure Induced by an Outer Measure......Page 357
17.4 The Construction of Outer Measures......Page 360
17.5 The Caratheodory-Hahn Theorem: The Extension of a Premeasure to a Measure......Page 363
18.1 Measurable Functions......Page 370
18.2 Integration of Nonnegative Measurable Functions......Page 376
18.3 Integration of General Measurable Functions......Page 383
18.4 The Radon-Nikodym Theorem......Page 392
18.5 The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem......Page 399
19.1 The Completeness of L^P(X, \mu,1 < p <=\infty......Page 405
19.2 The Riesz Representation Theorem for the Dual of L^P(X, \mu),1 < p<=\infty......Page 410
19.3 The Kantorovitch Representation Theorem for the Dual of L^\infty(X, \mu)......Page 415
19.4 Weak Sequential Compactness in L^P(X, \mu),1 < p < 1......Page 418
19.5 Weak Sequential Compactness in L^1(X, \mu) The Dunford-Pettis Theorem......Page 420
20.1 Product Measures: The Theorems of Fubini and Tonelli......Page 425
20.2 Lebesgue Measure on Euclidean Space R^n......Page 435
20.3 Cumulative Distribution Functions and Borel Measures on R......Page 448
20.4 Caratheodory Outer Measures and Hausdorff Measures on a Metric Space......Page 452
21 Measure and Topology......Page 457
21.1 Locally Compact Topological Spaces......Page 458
21.2 Separating Sets and Extending Functions......Page 463
21.3 The Construction of Radon Measures......Page 465
21.4 The Representation of Positive Linear Functionals on C_c(X): The Riesz-Markov Theorem......Page 468
21.5 The Riesz Representation Theorem for the Dual of C(X)......Page 473
21.6 Regularity Properties of Baire Measures......Page 481
22.1 Topological Groups: The General Linear Group......Page 488
22.2 Kakutani's Fixed Point Theorem......Page 491
22.3 Invariant Borel Measures on Compact Groups: von Neumann's Theorem......Page 496
22.4 Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov Theorem......Page 499
Bibliography......Page 506
Index......Page 508