Real analysis

Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson9780134588865, 013458886X

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.

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Table of contents :
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