《实分析（第三版）》课后习题答案

本书是一本优秀的教材，主要分三部分：第一部分为实变函数论，第二部分为抽象空间，第三部分为一般测度与积分论。书中不仅包含数学定理和定义，而且还提出了挑战性的问题，以便读者更深入地理解书中的内容。本书的题材是数学教学的共同基础，包含许多数学家的研究成果。

《实分析》（英文版第3版）是一本优秀的教材，主要分三部分：第一部分为实变函数论，第二部分为抽象空间，第三部分为一般测度与积分论。书中不仅包含数学定理和定义，而且还提出了挑战性的问题，以便读者更深入地理解书中的内容。《实分析》（英文版第3版）的题材是数学教学的共同基础，包含许多数学家的研究成果。

目录

• Prologue to the Student 1
• I Set Theory 6
• 1 Introduction 6
• 2 Functions 9
• 3 Unions, intersections, and complements 12
• 4 Algebras of sets 17
• 5 The axiom of choice and infinite direct products 19
• 6 Countable sets 20
• 7 Relations and equivalences 23
• 8 Partial orderings and the maximal principle 24
• 9 Well ordering and the countable ordinals 26
• Part One
• THEORY OF FUNCTIONS OF A
• REAL VARIABLE
• 2 The Real Number System 31
• 1 Axioms for the real numbers 31
• 2 The natural and rational numbers as subsets of R 34
• 3 The extended real numbers 36
• 4 Sequences of real numbers 37
• 5 Open and closed sets of real numbers 40
• 6 Continuous functions 47
• 7 Borel sets 52
• 3 Lebesgue Measure 54
• I Introduction 54
• 2 Outer measure 56
• 3 Measurable sets and Lebesgue measure 58
• *4 A nonmeasurable set 64
• 5 Measurable functions 66
• 6 Littlewood's three principles 72
• 4 The Lebesgue Integral 75
• 1 The Riemann integral 75
• 2 The Lebesgue integral of a bounded function over a set of finite
• measure 77
• 3 The integral of a nonnegative function 85
• 4 The general Lebesgue integral 89
• *5 Convergence in measure 95
• S Differentiation and Integration 97
• 1 Differentiation of monotone functions 97
• 2 Functions of bounded variation 102
• 3 Differentiation of an integral 104
• 4 Absolute continuity 108
• 5 Convex functions 113
• 6 The Classical Banach Spaces 118
• 1 The Lp spaces 118
• 2 The Minkowski and Holder inequalities 119
• 3 Convergence and completeness 123
• 4 Approximation in Lp 127
• 5 Bounded linear functionals on the Lp spaces 130
• Part Two
• ABSTRACT SPACES
• 7 Metric Spaces 139
• 1 Introduction 139
• 2 Open and closed sets 141
• 3 Continuous functions and homeomorphisms 144
• 4 Convergence and completeness 146
• 5 Uniform continuity and uniformity 148
• 6 Subspaces 151
• 7 Compact metric spaces 152
• 8 Baire category 158
• 9 Absolute Gs 164
• 10 The Ascoli-Arzela Theorem 167
• 8 Topological Spaces ltl
• I Fundamental notions 171
• 2 Bases and countability 175
• 3 The separation axioms and continuous real-valued
• functions 178
• 4 Connectedness 182
• 5 Products and direct unions of topological spaces 184
• *6 Topological and uniform properties 187
• *7 Nets 188
• 9 Compact and Locally Compact Spaces 190
• I Compact spaces 190
• 2 Countable compactness and the Bolzano-Weierstrass
• property 193
• 3 Products of compact spaces 196
• 4 Locally compact spaces 199
• 5 a-compact spaces 203
• *6 Paracompact spaces 204
• 7 Manifolds 206
• *8 The Stone-Cech compactification 209
• 9 The Stone-Weierstrass Theorem 210
• 10 Banach Spaces 217
• I Introduction 217
• 2 Linear operators 220
• 3 Linear functionals and the Hahn-Banach Theorem 222
• 4 The Closed Graph Theorem 224
• 5 Topological vector spaces 233
• 6 Weak topologies 236
• 7 Convexity 239
• 8 Hilbert space 245
• Part Three
• GENERAL MEASURE AND INTEGRATION
• THEORY
• 11 Measure and Integration 253
• 1 Measure spaces 253
• 2 Measurable functions 259
• 3 Integration 263
• 4 General Convergence Theorems 268
• 5 Signed measures 270
• 6 The Radon-Nikodym Theorem 276
• 7 The Lp-spaces 282
• 12 Measure and Outer Measure 288
• 1 Outer measure and measurability 288
• 2 The Extension Theorem 291
• 3 The Lebesgue-Stieltjes integral 299
• 4 Product measures 303
• 5 Integral operators 313
• *6 Inner measure 317
• *7 Extension by sets of measure zero 325
• 8 Caratheodory outer measure 326
• 9 Hausdorff measure 329
• 13 Measure and Topology 331
• 1 Baire sets and Borel sets 331
• 2 The regularity of Baire and Borel measures 337
• 3 The construction of Borel measures 345
• 4 Positive linear functionals and Borel measures 352
• 5 Bounded linear functionals on C(X) 355
• 14 Invariant Measures 361
• 1 Homogeneous spaces 361
• 2 Topological equicontinuity 362
• 3 The existence ofinvariant measures 365
• 4 Topological groups 370
• 5 Group actions and quotient spaces 376
• 6 Unicity ofinvariant measures 378
• 7 Groups ofdiffeomorphisms 388
• 15 Mappings of Measure Spaces 392
• 1 Point mappings and set mappings 392
• 2 Boolean algebras 394
• 3 Measure algebras 398
• 4 Borel equivalences 401
• 5 Borel measures on complete separable metric spaces 406
• 6 Set mappings and point mappings on complete separable
• metric spaces 412
• 7 The isometries of Lp 415
• 16 The Daniell Integral 419
• 1 Introduction 419
• 2 The Extension Theorem 422
• 3 Uniqueness 427
• 4 Measurability and measure 429
• Bibliography 435
• Index of Symbols 437
• Subject Index 439