- 作 者：（美）Frank Jones
- 出 版 社：北京：世界图书出版公司北京公司
- 出版年份：2010
- ISBN：9787510005558
- 标注页数：592 页
- PDF页数：607 页
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1 Introduction to Rn 1
A Sets 1
B Countable Sets 4
C Topology 5
D Compact Sets 10
E Continuity 15
F The Distance Function 20
2 Lebesgue Measure on Rn 25
A Construction 25
B Properties of Lebesgue Measure 49
C Appendix:Proof of P1 and P2 60
3 Invariance of Lebesgue Measure 65
A Some Linear Algebra 66
B Translation and Dilation 71
C Orthogonal Matrices 73
D The General Matrix 75
4 Some Interesting Sets 81
A A Nonmeasurable Set 81
B A Bevy of Cantor Sets 83
C The Lebesgue Function 86
D Appendix:The Modulus of Continuity of the Lebesgue Functions 95
5 Algebras of Sets and Measurable Functions 103
A Algebras and σ-Algebras 103
B Borel Sets 107
C A Measurable Set which Is Not a Borel Set 110
D Measurable Functions 112
E Simple Functions 117
6 Integration 121
A Nonnegative Functions 121
B General Measurable Functions 130
C Almost Everywhere 135
D Integration Over Subsets of Rn 139
E Generalization:Measure Spaces 142
F Some Calculations 147
G Miscellany 152
7 Lebesgue Integral on Rn 157
A Riemann Integral 157
B Linear Change of Variables 170
C Approximation of Functions in L1 171
D Continuity of Translation in L1 180
8 Fubini's Theorem for Rn 181
9 The Gamma Function 199
A Definition and Simple Properties 199
B Generalization 202
C The Measure of Balls 205
D Further Properties of the Gamma Function 209
E Stirling's Formula 212
F The Gamma Function on R 216
10 Lp Spaces 221
A Definition and Basic Inequalities 221
B Metric Spaces and Normed Spaces 227
C Completeness of Lp 231
D The Case p＝∞ 235
E Relations between Lp Spaces 238
F Approximation by C∞c(Rn) 244
G Miscellaneous Problems 246
H The Case 0<p<1 250
11 Products of Abstract Measures 255
A Products of σ-Algebras 255
B Monotone Classes 258
C Construction of the Product Measure 261
D The Fubini Theorem 268
E The Generalized Minkowski Inequality 271
12 Convolutions 277
A Formal Properties 277
B Basic Inequalities 280
C Approximate Identities 284
13 Fourier Transform on Rn 293
A Fourier Transform of Functions in L1(Rn) 293
B The Inversion Theorem 308
C The Schwartz Class 320
D The Fourier-Plancherel Transform 323
E Hilbert Space 334
F Formal Application to Differential Equations 339
G Bessel Functions 344
H Special Results for n=1 352
I Hermite Polynomials 356
14 Fourier Series in One Variable 367
A Periodic Functions 367
B Trigonometric Series 373
C Fourier Coefficients 392
D Convergence of Fourier Series 400
E Summability of Fourier Series 410
F A Counterexample 418
G Parseval's Identity 421
H Poisson Summation Formula 428
I A Special Class of Sine Series 436
15 Differentiation 447
A The Vitali Covering Theorem 448
B The Hardy-Littlewood Maximal Function 450
C Lebesgue's Differentiation Theorem 456
D The Lebesgue Set of a Function 458
E Points of Density 463
F Applications 466
G The Vitali Covering Theorem(Again) 478
H The Besicovitch Covering Theorem 482
I The Lebesgue Set of Order p 491
J Change of Variables 494
K Noninvertible Mappings 505
16 Differentiation for Functions on R 511
A Monotone Functions 511
B Jump Functions 521
C Another Theorem of Fubini 527
D Bounded Variation 530
E Absolute Continuity 544
F Further Discussion of Absolute Continuity 553
G Arc Length 563
H Nowhere Differentiable Functions 570
I Convex Functions 576
Index 581
Symbol Index 587