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 102 Regular Outer Measures......Page 105 Nonmeasurable Sets......Page 109 More About Method I......Page 112 Completions......Page 116 Additional Problems for Chapter 2......Page 118 Metric Outer Measures......Page 122 Metric Space......Page 123 Metric Outer Measures......Page 126 Method II......Page 131 Approximations......Page 135 Construction of Lebesgue--Stieltjes Measures......Page 137 Properties of Lebesgue--Stieltjes Measures......Page 143 Lebesgue--Stieltjes Measures in IRn......Page 148 Hausdorff Measures and Hausdorff Dimension......Page 150 Methods III and IV......Page 157 Additional Remarks......Page 162 Additional Problems for Chapter 3......Page 166 Measurable Functions......Page 171 Definitions and Basic Properties......Page 172 Sequences of Measurable Functions......Page 177 Egoroff's Theorem......Page 182 Approximations by Simple Functions......Page 185 Approximation by Continuous Functions......Page 189 Additional Problems for Chapter 4......Page 194 Integration......Page 198 Introduction......Page 199 Integrals of Nonnegative Functions......Page 203 Fatou's Lemma......Page 207 Integrable Functions......Page 211 Riemann and Lebesgue......Page 215 Countable Additivity of the Integral......Page 223 Absolute Continuity......Page 226 Radon--Nikodym Theorem......Page 231 Convergence Theorems......Page 238 Relations to Other Integrals......Page 245 Integration of Complex Functions......Page 249 Additional Problems for Chapter 5......Page 253 Fubini's Theorem......Page 258 Product Measures......Page 259 Fubini's Theorem......Page 267 Tonelli's Theorem......Page 269 Additional Problems for Chapter 6......Page 271 The Vitali Covering Theorem......Page 274 Functions of Bounded Variation......Page 280 The Banach--Zarecki Theorem......Page 284 Determining a Function by Its Derivative......Page 287 Calculating a Function from Its Derivative......Page 289 Total Variation of a Continuous Function......Page 296 VBG* Functions......Page 302 Approximate Continuity, Lebesgue Points......Page 306 Additional Problems for Chapter 7......Page 312 Differentiation of Measures......Page 319 Differentiation of Lebesgue--Stieltjes Measures......Page 320 The Cube Basis; Ordinary Differentiation......Page 324 The Lebesgue Decomposition Theorem......Page 330 The Interval Basis; Strong Differentiation......Page 332 Net Structures......Page 339 Radon--Nikodym Derivative in a Measure Space......Page 345 Summary, Comments, and References......Page 353 Additional Problems for Chapter 8......Page 356 Definitions and Examples......Page 358 Convergence and Related Notions......Page 367 Continuity......Page 370 Homeomorphisms and Isometries......Page 374 Separable Spaces......Page 378 Complete Spaces......Page 380 Contraction Maps......Page 385 Applications of Contraction Mappings......Page 387 Compactness......Page 393 Totally Bounded Spaces......Page 397 Compact Sets in C(X)......Page 398 Application of the Arzelà--Ascoli Theorem......Page 402 The Stone--Weierstrass Theorem......Page 404 The Isoperimetric Problem......Page 407 More on Convergence......Page 410 Additional Problems for Chapter 9......Page 414 The Baire Category Theorem......Page 417 The Banach--Mazur Game......Page 423 The First Classes of Baire and Borel......Page 428 Properties of Baire-1 Functions......Page 433 Topologically Complete Spaces......Page 437 Applications to Function Spaces......Page 441 Additional Problems for Chapter 10......Page 452 Analytic Sets......Page 458 Products of Metric Spaces......Page 459 Baire Space......Page 460 Analytic Sets......Page 463 Borel Sets......Page 467 An Analytic Set That Is Not Borel......Page 471 Measurability of Analytic Sets......Page 473 The Suslin Operation......Page 475 A Method to Show a Set Is Not Borel......Page 477 Differentiable Functions......Page 480 Additional Problems for Chapter 11......Page 484 Normed Linear Spaces......Page 487 Compactness......Page 493 Linear Operators......Page 497 Banach Algebras......Page 501 The Hahn--Banach Theorem......Page 504 Improving Lebesgue Measure......Page 508 The Dual Space......Page 514 The Riesz Representation Theorem......Page 517 Separation of Convex Sets......Page 523 An Embedding Theorem......Page 528 The Uniform Boundedness Principle......Page 530 An Application to Summability......Page 533 The Open Mapping Theorem......Page 537 The Closed Graph Theorem......Page 541 Additional Problems for Chapter 12......Page 543 The Basic Inequalities......Page 546 The p and Lp Spaces (1p< )......Page 550 The Spaces and L......Page 553 Separability......Page 555 The Spaces 2 and L2......Page 557 Continuous Linear Functionals......Page 563 The Lp Spaces (0