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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