Real Analysis

This book provides an introductory chapter containing background material as well as a mini-overview of much of the course, making the book accessible to readers with varied backgrounds. It uses a wealth of examples to introduce topics and to illustrate important concepts. KEY TOPICS: Explains the ideas behind developments and proofs -- showing that proofs come not from “ magical methods” but from natural processes. Introduces concepts in stages, and features applications of abstract theorems to concrete settings -- showing the power of an abstract approach in problem solving. Preface......Page 3 Background and Preview......Page 12 The Real Numbers......Page 13 Compact Sets of Real Numbers......Page 18 Countable Sets......Page 21 Uncountable Cardinals......Page 24 Transfinite Ordinals......Page 27 Category......Page 30 Outer Measure and Outer Content......Page 33 Small Sets......Page 35 Measurable Sets of Real Numbers......Page 38 Nonmeasurable Sets......Page 42 Zorn's Lemma......Page 45 Borel Sets of Real Numbers......Page 47 Analytic Sets of Real Numbers......Page 49 Bounded Variation......Page 51 Newton's Integral......Page 54 Cauchy's Integral......Page 55 Riemann's Integral......Page 57 Volterra's Example......Page 60 Riemann--Stieltjes Integral......Page 62 Lebesgue's Integral......Page 65 The Generalized Riemann Integral......Page 67 Additional Problems for Chapter 1......Page 70 Measure Spaces......Page 74 One-Dimensional Lebesgue Measure......Page 75 Additive Set Functions......Page 80 Measures and Signed Measures......Page 86 Limit Theorems......Page 89 Jordan and Hahn Decomposition......Page 93 Complete Measures......Page 96 Outer Measures......Page 99 Method I......Page 103 Regular Outer Measures......Page 106 Nonmeasurable Sets......Page 110 More About Method I......Page 113 Completions......Page 116 Additional Problems for Chapter 2......Page 119 Metric Outer Measures......Page 123 Metric Space......Page 124 Metric Outer Measures......Page 127 Method II......Page 132 Approximations......Page 136 Construction of Lebesgue--Stieltjes Measures......Page 138 Properties of Lebesgue--Stieltjes Measures......Page 144 Lebesgue--Stieltjes Measures in IRn......Page 149 Hausdorff Measures and Hausdorff Dimension......Page 151 Methods III and IV......Page 158 Additional Remarks......Page 163 Additional Problems for Chapter 3......Page 167 Measurable Functions......Page 172 Definitions and Basic Properties......Page 173 Sequences of Measurable Functions......Page 178 Egoroff's Theorem......Page 183 Approximations by Simple Functions......Page 186 Approximation by Continuous Functions......Page 190 Additional Problems for Chapter 4......Page 195 Integration......Page 199 Introduction......Page 200 Integrals of Nonnegative Functions......Page 204 Fatou's Lemma......Page 208 Integrable Functions......Page 212 Riemann and Lebesgue......Page 216 Countable Additivity of the Integral......Page 224 Absolute Continuity......Page 227 Radon--Nikodym Theorem......Page 232 Convergence Theorems......Page 239 Relations to Other Integrals......Page 246 Integration of Complex Functions......Page 250 Additional Problems for Chapter 5......Page 254 Fubini's Theorem......Page 259 Product Measures......Page 260 Fubini's Theorem......Page 268 Tonelli's Theorem......Page 270 Additional Problems for Chapter 6......Page 272 The Vitali Covering Theorem......Page 275 Functions of Bounded Variation......Page 281 The Banach--Zarecki Theorem......Page 285 Determining a Function by Its Derivative......Page 288 Calculating a Function from Its Derivative......Page 290 Total Variation of a Continuous Function......Page 297 VBG* Functions......Page 303 Approximate Continuity, Lebesgue Points......Page 307 Additional Problems for Chapter 7......Page 313 Differentiation of Measures......Page 320 Differentiation of Lebesgue--Stieltjes Measures......Page 321 The Cube Basis; Ordinary Differentiation......Page 325 The Lebesgue Decomposition Theorem......Page 331 The Interval Basis; Strong Differentiation......Page 333 Net Structures......Page 340 Radon--Nikodym Derivative in a Measure Space......Page 346 Summary, Comments, and References......Page 354 Additional Problems for Chapter 8......Page 357 Definitions and Examples......Page 359 Convergence and Related Notions......Page 368 Continuity......Page 371 Homeomorphisms and Isometries......Page 375 Separable Spaces......Page 379 Complete Spaces......Page 381 Contraction Maps......Page 386 Applications of Contraction Mappings......Page 388 Compactness......Page 394 Totally Bounded Spaces......Page 398 Compact Sets in C(X)......Page 399 Application of the Arzelà--Ascoli Theorem......Page 403 The Stone--Weierstrass Theorem......Page 405 The Isoperimetric Problem......Page 408 More on Convergence......Page 411 Additional Problems for Chapter 9......Page 415 The Baire Category Theorem......Page 418 The Banach--Mazur Game......Page 424 The First Classes of Baire and Borel......Page 429 Properties of Baire-1 Functions......Page 434 Topologically Complete Spaces......Page 438 Applications to Function Spaces......Page 442 Additional Problems for Chapter 10......Page 453 Analytic Sets......Page 459 Products of Metric Spaces......Page 460 Baire Space......Page 461 Analytic Sets......Page 464 Borel Sets......Page 468 An Analytic Set That Is Not Borel......Page 472 Measurability of Analytic Sets......Page 474 The Suslin Operation......Page 476 A Method to Show a Set Is Not Borel......Page 478 Differentiable Functions......Page 481 Additional Problems for Chapter 11......Page 485 Normed Linear Spaces......Page 488 Compactness......Page 494 Linear Operators......Page 498 Banach Algebras......Page 502 The Hahn--Banach Theorem......Page 505 Improving Lebesgue Measure......Page 509 The Dual Space......Page 515 The Riesz Representation Theorem......Page 518 Separation of Convex Sets......Page 524 An Embedding Theorem......Page 529 The Uniform Boundedness Principle......Page 531 An Application to Summability......Page 534 The Open Mapping Theorem......Page 538 The Closed Graph Theorem......Page 542 Additional Problems for Chapter 12......Page 544 The Basic Inequalities......Page 547 The p and Lp Spaces (1p< )......Page 551 The Spaces and L......Page 554 Separability......Page 556 The Spaces 2 and L2......Page 558 Continuous Linear Functionals......Page 564 The Lp Spaces (0