简介
本书为数学与统计学专业研究生实分析课程的基础教材，1963年出版了第1版，1987年修订的第3版在前两版的基础上进行了改写和补充，增加了可变测度一章。在过去的40我年中，本书已被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。
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。
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
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
Real analysis = 实分析 / 3rd ed.
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