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欧氏空间上的勒贝格积分 修订版 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

欧氏空间上的勒贝格积分 修订版 英文版
  • （美）Frank Jones 著
  • 出版社： 北京：世界图书出版公司北京公司
  • ISBN：9787510005558
  • 出版时间：2010
  • 标注页数：592页
  • 文件大小：114MB
  • 文件页数：607页
  • 主题词：欧几里德空间－勒贝格积分－英文

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图书目录

1　Introduction to Rn1

A　Sets1

B　Countable Sets4

C　Topology5

D　Compact Sets10

E　Continuity15

F　The Distance Function20

2　Lebesgue Measure on Rn25

A　Construction25

B　Properties of Lebesgue Measure49

C　Appendix:Proof of P1 and P260

3　Invariance of Lebesgue Measure65

A　Some Linear Algebra66

B　Translation and Dilation71

C　Orthogonal Matrices73

D　The General Matrix75

4　Some Interesting Sets81

A　A Nonmeasurable Set81

B　A Bevy of Cantor Sets83

C　The Lebesgue Function86

D　Appendix:The Modulus of Continuity of the Lebesgue Functions95

5　Algebras of Sets and Measurable Functions103

A　Algebras and σ-Algebras103

B　Borel Sets107

C　A Measurable Set which Is Not a Borel Set110

D　Measurable Functions112

E　Simple Functions117

6　Integration121

A　Nonnegative Functions121

B　General Measurable Functions130

C　Almost Everywhere135

D　Integration Over Subsets of Rn139

E　Generalization:Measure Spaces142

F　Some Calculations147

G　Miscellany152

7　Lebesgue Integral on Rn157

A　Riemann Integral157

B　Linear Change of Variables170

C　Approximation of Functions in L1171

D　Continuity of Translation in L1180

8　Fubini's Theorem for Rn181

9　The Gamma Function199

A　Definition and Simple Properties199

B　Generalization202

C　The Measure of Balls205

D　Further Properties of the Gamma Function209

E　Stirling's Formula212

F　The Gamma Function on R216

10　Lp Spaces221

A　Definition and Basic Inequalities221

B　Metric Spaces and Normed Spaces227

C　Completeness of Lp231

D　The Case p＝∞235

E　Relations between Lp Spaces238

F　Approximation by C∞c(Rn)244

G　Miscellaneous Problems246

H　The Case 0<p<1250

11　Products of Abstract Measures255

A　Products of σ-Algebras255

B　Monotone Classes258

C　Construction of the Product Measure261

D　The Fubini Theorem268

E　The Generalized Minkowski Inequality271

12　Convolutions277

A　Formal Properties277

B　Basic Inequalities280

C　Approximate Identities284

13　Fourier Transform on Rn293

A　Fourier Transform of Functions in L1(Rn)293

B　The Inversion Theorem308

C　The Schwartz Class320

D　The Fourier-Plancherel Transform323

E　Hilbert Space334

F　Formal Application to Differential Equations339

G　Bessel Functions344

H　Special Results for n=1352

I　Hermite Polynomials356

14　Fourier Series in One Variable367

A　Periodic Functions367

B　Trigonometric Series373

C　Fourier Coefficients392

D　Convergence of Fourier Series400

E　Summability of Fourier Series410

F　A Counterexample418

G　Parseval's Identity421

H　Poisson Summation Formula428

I　A Special Class of Sine Series436

15　Differentiation447

A　The Vitali Covering Theorem448

B　The Hardy-Littlewood Maximal Function450

C　Lebesgue's Differentiation Theorem456

D　The Lebesgue Set of a Function458

E　Points of Density463

F　Applications466

G　The Vitali Covering Theorem(Again)478

H　The Besicovitch Covering Theorem482

I　The Lebesgue Set of Order p491

J　Change of Variables494

K　Noninvertible Mappings505

16　Differentiation for Functions on R511

A　Monotone Functions511

B　Jump Functions521

C　Another Theorem of Fubini527

D　Bounded Variation530

E　Absolute Continuity544

F　Further Discussion of Absolute Continuity553

G　Arc Length563

H　Nowhere Differentiable Functions570

I　Convex Functions576

Index581

Symbol Index587

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