Real Analysis, Second Edition

Ben shu zhu yao fen san bu fen:di yi bu fen wei shi bian han shu lun, Di er bu fen wei chou xiang kong jian, Di san bu fen wei yi ban ce du yu ji fen lun. REAL ANALYSIS, 2ND ED.......Page 1 Title Page......Page 3 Preface......Page 4 Contents......Page 6 Prologue to the Student......Page 12 1 Introduction......Page 16 2 Functions......Page 19 3 Unions, Intersections, and Complements......Page 22 4 Algebras of Sets......Page 27 5 The Axiom of Choice and Infinite Direct Products......Page 29 6 Countable Sets......Page 30 7 Relations and Equivalences......Page 33 8 Partial Orderings and the Maximal Principle......Page 34 9 Well Ordering and the Countable Ordinals......Page 35 Part I: Theory of Functions of a Real Variable......Page 38 1 Axioms for the Real Numbers......Page 40 2 The Natural and Rational Numbers as Subsets of R......Page 43 3 The Extended Real Numbers......Page 45 4 Sequences of Real Numbers......Page 46 5 Open and Closed Sets of Real Numbers......Page 49 6 Continuous Functions......Page 55 7 Borel Sets......Page 61 1 Introduction......Page 63 2 Outer Measure......Page 65 3 Measurable Sets and Lebesgue Measure......Page 67 *4 A Nonmeasurable Set......Page 74 5 Measurable Functions......Page 76 6 Littlewood's Three Principles......Page 82 1 The Riemann Integral......Page 84 2 The Lebesgue Integral of a Bounded Function over a Set of Finite Measure......Page 86 3 The Integral of a Nonnegative Function......Page 93 4 The General Lebesgue Integral......Page 97 *5 Convergence in Measure......Page 102 1 Differentiation of Monotone Functions......Page 105 2 Functions of Bounded Variation......Page 109 3 Differentiation of an Integral......Page 112 4 Absolute Continuity......Page 115 *5 Convex Functions......Page 119 1 The L[sup(p)] Spaces......Page 122 2 The Hölder and Minkowski Inequalities......Page 123 3 Convergence and Completeness......Page 126 4 Bounded Linear Functionals on the L[sup(p)] Spaces......Page 130 Part II: Abstract Spaces......Page 136 1 Introduction......Page 138 2 Open and Closed Sets......Page 140 3 Continuous Functions and Homeomorphisms......Page 142 4 Convergence and Completeness......Page 144 5 Uniform Continuity and Uniformity......Page 146 6 Subspaces......Page 148 7 Baire Category......Page 150 1 Fundamental Notions......Page 153 2 Bases and Countability......Page 156 3 The Separation Axioms and Continuous Real-Valued Functions......Page 158 4 Product Spaces......Page 161 5 Connectedness......Page 163 *6 Absolute G[sub(δ)]'s......Page 165 *7 Nets......Page 166 1 Basic Properties......Page 168 2 Countable Compactness and the Bolzano–Weierstrass Property......Page 170 3 Compact Metric Spaces......Page 174 4 Products of Compact Spaces......Page 176 5 Locally Compact Spaces......Page 179 *6 The Stone–Čech Compactification......Page 181 7 The Stone–Weierstrass Theorem......Page 182 *8 The Ascoli Theorem......Page 188 1 Introduction......Page 192 2 Linear Operators......Page 195 3 Linear Functionals and the Hahn–Banach Theorem......Page 197 4 The Closed Graph Theorem......Page 204 *5 Topological Vector Spaces......Page 208 *6 Weak Topologies......Page 211 *7 Convexity......Page 214 8 Hilbert Space......Page 221 Part III: General Measure and Integration Theory......Page 226 1 Measure Spaces......Page 228 2 Measurable Functions......Page 234 3 Integration......Page 236 *4 General Convergence Theorems......Page 242 5 Signed Measures......Page 243 6 The Radon–Nikodym Theorem......Page 249 7 The L[sup(p)] Spaces......Page 254 1 Outer Measure and Measurability......Page 261 2 The Extension Theorem......Page 264 *3 The Lebesgue–Stieltjes Integral......Page 272 4 Product Measures......Page 275 *5 Inner Measure......Page 285 *6 Extension by Sets of Measure Zero......Page 292 *7 Carathéodory Outer Measure......Page 294 1 Introduction......Page 297 2 The Extension Theorem......Page 299 3 Uniqueness......Page 305 4 Measurability and Measure......Page 306 1 Baire Sets and Borel Sets......Page 312 2 Positive Linear Functionals and Baire Measures......Page 315 3 Bounded Linear Functionals on C(X)......Page 319 *4 The Borel Extension of a Measure......Page 324 1 Point Mappings and Set Mappings......Page 328 2 Measure Algebras......Page 330 3 Borel Equivalences......Page 335 4 Set Mappings and Point Mappings on Complete Metric Spaces......Page 339 5 The Isometries of L[sup(p)]......Page 342 Epilogue......Page 346 Bibliography......Page 348 Index of Symbols......Page 350 Subject Index......Page 352