实分析（英文版·第4版）

2.7 The Cantor Set and the Cantor-Lebesgue Function 49
Lebesgue Measurable Functions 54
3.1 Sums,Products,andCompositions 54
3.2 Sequential Pointwise Limits and Simple Approximation 60
3.3 Littlewood’s Three Principles, Egoroff’s Theorem, and Lusin’s Theorem 64
Lebesgue Integration 68
4.1 TheRiemannIntegral 68
4.2 The Lebesgue Integral of a Bounded Measurable Function over a Set of
FiniteMeasure 71
4.3 The Lebesgue Integral of a Measurable Nonnegative Function 79
4.4 TheGeneralLebesgueIntegral 85
4.5 Countable Additivity and Continuity of Integration 90
4.6 Uniform Integrability: The Vitali Convergence Theorem 92
Lebesgue Integration: Further Topics 97
5.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem 97
5.2 ConvergenceinMeasure 99
5.3 Characterizations of Riemann and Lebesgue Integrability 102
Differentiation and Integration 107
6.1 ContinuityofMonotoneFunctions 108
6.2 Differentiability of Monotone Functions: Lebesgue’s Theorem 109
6.3 Functions of Bounded Variation: Jordan’s Theorem 116
6.4 AbsolutelyContinuousFunctions . 119
6.5 Integrating Derivatives: Differentiating Indefinite Integrals . 124
6.6 ConvexFunctions . 130
7The Lp Spaces: Completeness and Approximation 135
7.1 NormedLinearSpaces . 135
7.2 The Inequalities of Young, H older, and Minkowski 139¨
7.3 Lp IsComplete:TheRiesz-FischerTheorem 144
7.4 ApproximationandSeparability 150
8The Lp Spaces: Duality and Weak Convergence 155
8.1 The Riesz Representation for the Dual of Lp, 1 155
8.2 Weak Sequential Convergence in Lp 162
8.3 WeakSequentialCompactness 171
8.4 TheMinimizationofConvexFunctionals174
II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces 181
Metric Spaces: General Properties 183
9.1 ExamplesofMetricSpaces 183
9.2 Open Sets, Closed Sets, and Convergent Sequences 187
9.3 ContinuousMappingsBetweenMetricSpaces 190
9.4 CompleteMetricSpaces 193
9.5 CompactMetricSpaces . 197
9.6 SeparableMetricSpaces 204
10 Metric Spaces: Three Fundamental Theorems 206
10.1TheArzela-AscoliTheorem `. 206
10.2TheBaireCategoryTheorem 211
10.3TheBanachContractionPrinciple. 215
11 Topological Spaces: General Properties 222
11.1 OpenSets,ClosedSets,Bases,andSubbases. 222
11.2TheSeparationProperties 227
11.3CountabilityandSeparability 228
11.4 Continuous Mappings Between Topological Spaces 230
11.5CompactTopologicalSpaces. 233
11.6ConnectedTopologicalSpaces 237
12 Topological Spaces: Three Fundamental Theorems 239
12.1 Urysohn’s Lemma and the Tietze Extension Theorem . 239
12.2TheTychonoffProductTheorem . 244
12.3TheStone-WeierstrassTheorem 247
13 Continuous Linear Operators Between Banach Spaces 253
13.1NormedLinearSpaces . 253
13.2LinearOperators . 256
13.3 Compactness Lost: Infinite Dimensional Normed Linear Spaces 259
13.4 TheOpenMappingandClosedGraphTheorems . 263
13.5TheUniformBoundednessPrinciple 268
14 Duality for Normed Linear Spaces 271
14.1 Linear Functionals, Bounded Linear Functionals, and Weak Topologies 271
14.2TheHahn-BanachTheorem . 277
14.3 Reflexive Banach Spaces and Weak Sequential Convergence 282
14.4 LocallyConvexTopologicalVectorSpaces 286
14.5 The Separation of Convex Sets and Mazur’s Theorem . 290
14.6TheKrein-MilmanTheorem. 295
15 Compactness Regained: The Weak Topology 298
15.1 Alaoglu’sExtensionofHelley’sTheorem . 298
15.2 Reflexivity and Weak Compactness: Kakutani’s Theorem 300
15.3 Compactness and Weak Sequential Compactness: The Eberlein-ˇ
Smulian Theorem 302
15.4MetrizabilityofWeakTopologies . 305
16 Continuous Linear Operators on Hilbert Spaces 308
16.1TheInnerProductandOrthogonality 309
16.2 The Dual Space and Weak Sequential Convergence 313
16.3 Bessel’sInequalityandOrthonormalBases . 316
16.4 AdjointsandSymmetryforLinearOperators 319
16.5CompactOperators 324
16.6TheHilbert-SchmidtTheorem 326
16.7 The Riesz-Schauder Theorem: Characterization of Fredholm Operators 329
III Measure and Integration: General Theory 335
17 General Measure Spaces: Their Properties and Construction 337
17.1MeasuresandMeasurableSets 337
17.2 Signed Measures: The Hahn and Jordan Decompositions 342
17.3 The Carath′346
eodory Measure Induced by an Outer Measure
17.4TheConstructionofOuterMeasures 349
17.5 The Carath′eodory-Hahn Theorem: The Extension of a Premeasure to a
Measure 352
18 Integration Over General Measure Spaces 359
18.1MeasurableFunctions 359
18.2 Integration of Nonnegative Measurable Functions 365
18.3 Integration of General Measurable Functions 372
18.4TheRadon-NikodymTheorem 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 LpX,μ1 ≤≤. 394
19.2 The Riesz Representation Theorem for the Dual of LpX,μ1 ≤≤ 399
19.3 The Kantorovitch Representation Theorem for the Dual of L∞X,μ. 404
19.4 Weak Sequential Compactness in LpX,μ1 [p[ 1. 407
19.5 Weak Sequential Compactness in L1X,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 437
20.4 Caratheodory Outer Measures and Hausdorff Measures on a Metric Space ′. 441
21 Measure and Topology 446
21.1LocallyCompactTopologicalSpaces 447
21.2 SeparatingSetsandExtendingFunctions452
21.3TheConstructionofRadonMeasures 454
21.4 The Representation of Positive Linear Functionals on CcX:The Riesz-
MarkovTheorem . 457
21.5 The Riesz Representation Theorem for the Dual of CX 462
21.6 RegularityPropertiesofBaireMeasures 470
22 Invariant Measures 477
22.1 Topological Groups: The General Linear Group . 477
22.2Kakutani’sFixedPointTheorem . 480
22.3 Invariant Borel Measures on Compact Groups: von Neumann’s Theorem 485
22.4 Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov
Theorem 488
Bibliography 495
Index 497

.　　The bibliography lists a number of books that are not specifically referenced but should be consulted for supplementary material and different viewpoints. In particular, two books on the interesting history of mathematical analysis are listed.
　　SUGGESTIONS FOR COURSES: FIRST SEMESTER
　　In Chapter 1, all the background elementary analysis and topology of the real line needed for Part I is established. This initial chapter is meant to be a handy reference. Core material comprises Chapters 2, 3, and 4, the first five sections of Chapter 6, Chapter 7, and the first section of Chapter 8. Following this, selections can be made: Sections 8.2-8.4 are interesting for students who will continue to study duality and compactness for normed linear spaces, while Section 5.3 contains two jewels of classical analysis, the characterization of Lebesgue integrability and of Riemann integrability for bounded functions.
　　SUGGESTIONS FOR COURSES: SECOND SEMESTER
　　This course should be based on Part III. Initial core material comprises Section 17.1, Section 18.1-18.4, and Sections 19.1-19.3. The remaining sections in Chapter 17 may be covered at the beginning or as they are needed later: Sections 17.3-17.5 before Chapter 20, and Section 17.2 before Chapter 21. Chapter 20 can then be covered. None of this material depends on Part II. Then several selected topics can be chosen, dipping into Part II as needed.
　　·Suggestion 1: Prove the Baire Category Theorem and its corollary regarding the partial continuity of the pointwise limit of a sequence of continuous functions (Theorem 7 of Chapter 10), infer from the Riesz-Fischer Theorem that the Nikodym metric space is complete (Theorem 23 of Chapter 18), prove the Vitali-Hahn-Saks Theorem and then prove the Dunford-Pettis Theorem.
　　·Suggestion 2: Cover Chapter 21 (omitting Section 20.5) on Measure and Topology, with the option of assuming the topological spaces are metrizable, so 20.1 can be skipped.
　　·Suggestion 3: Prove Riesz's Theorem regarding the closed unit ball of an infinite dimensional normed linear space being noncompact with respect to the topology induced by the norm. Use this as a motivation for regaining sequential compactness with respect to weaker topologies, then use Helley's Theorem to obtain weak sequential compactness properties of the LP(X, μ) spaces, 1 [ p [ ∞, if Lq(x, μ) is separable and, if Chapter 21 has already been covered, weak-, sequential compactness results for Radon measures on the Borel σ-algebra of a compact metric space.
　　SUGGESTIONS FOR COURSES: THIRD SEMESTER
　　I have used Part II, with some supplemental material, for a course on functional analysis, for students who had taken the first two semesters; the material is tailored, of course, to that chosen for the second semester. Chapter 16 on bounded linear operators on a Hilbert space may be covered right after Chapter 13 on bounded linear operators on a Banach space, since the results regarding weak sequential compactness are obtained directly from the existence of an orthogonal complement for each closed subspace of a Hilbert space. Part II should be interlaced with selections from Part III to provide applications of the abstract space theory to integration. For instance, reflexivity and weak compactness can be considered in general LP(X, μ) spaces, using material from Chapter 19. The above suggestion 1 for the second semester course can be taken in the third semester rather than the second, providing a truly striking application of the Baire Category Theorem. The establishment, in Chapter 21, of the representation of the dual of C(X), where X is a compact Hausdorff space, provides another collection of spaces, spaces of signed Radon measures, to which the theorems of Helley, Alaoghi, and Krein-Milman apply. By covering Chapter 22 on Invariant Measures, the student will encounter applications of Alaoglu's Theorem and the Krein-Milman Theorem to prove the existence of Haar measure on a compact group and the existence of measures with respect to which a mapping is ergodic (Theorem 14 of Chapter 22), and an application of Helley's Theorem to establish the existence of invariant measures (the Bogoliubov-Krilov Theorem).
　　I welcome comments at pmf@math.umd.edu. A list of errata and remarks will be placed on www.math.umd.edu/~pmf/RealAnalysis.
　　ACKNOWLEDGMENTS
　　It is a pleasure to acknowledge my indebtedness to teachers, colleagues, and students. A penultimate draft of the entire manuscript was read by Diogo Arsenio, whom I warmly thank for his observations and suggestions, which materially improved that draft. Here in my mathematical home, the University of Maryland, I have written notes for various analysis courses, which have been incorporated into the present edition. A number of students in my graduate analysis classes worked through parts of drafts of this edition, and their comments and suggestions have been most helpful: I thank Avner Halevy, Kevin McGoff, and Himanshu Tiagi. I am pleased to acknowledge a particular debt to Brendan Berg who created the index, proofread the final manuscript, and kindly refined my tex skills. I have benefited from conversations with many friends and colleagues; in particular, with Diogo Ars6nio, Michael Boyle, Michael Brin, Craig Evans, Manos Grillakis, Brian Hunt, Jacobo Pejsachowicz, Eric Slud, Robert Warner, and Jim Yorke. Publisher and reviewers: J. Thomas Beale, Duke University; Richard Carmichael, Wake Forest University; Michael Goldberg, Johns Hopkins University; Paul Joyce, University of Idaho; Dmitry Kaliuzhnyi-Verbovetskyi, Drexel University; Giovanni Leoni, Carnegie Mellon University; Bruce Mericle, Mankato State University; Stephen Robinson, Wake Forest University; Martin Schechter, University of California-Irvine; James Stephen White, Jacksonville State University; and Shanshuang Yang, Emory University.
　　Patrick M. Fitzpatrick
　　College Park, MD
　　November, 2009