分析 第3卷 英文PDF电子书下载

Chapter Ⅸ Elements of measure theory 3

1 Measurable spaces 3

σ-algebras 3

The Borel σ-algebra 5

The second countability axiom 6

Generating the Borel σ-algebra with intervals 8

Bases of topological spaces 9

The product topology 10

Product Borelσ-algebras 12

Measurability of sections 13

2 Measures 17

Set functions 17

Measure spaces 18

Properties of measures 18

Null sets 20

3 Outer measures 24

The construction of outer measures 24

The Lebesgue outer measure 25

The Lebesgue-Stieltjes outer measure 28

Hausdorff outer measures 29

4 Measurable sets 32

Motivation 32

Theσ-algebra of μ-measurable sets 33

Lebesgue measure and Hausdorff measure 35

Metric measures 36

5 The Lebesgue measure 40

The Lebesgue measure space 40

The Lebesgue measure is regular 41

A characterization of Lebesgue measurability 44

Images of Lebesgue measurable sets 44

The Lebesgue measure is translation invariant 47

A characterization of Lebesgue measure 48

The Lebesgue measure is invariant under rigid motions 50

The substitution rule for linear maps 51

Sets without Lebesgue measure 53

Chapter Ⅹ Integration theory 62

1 Measurable functions 62

Simple functions and measurable functions 62

A measurability criterion 64

Measurable ？-valued functions 67

The lattice of measurable ？-valued functions 68

Pointwise limits of measurable functions 73

Radon measures 74

2 Integrable functions 80

The integral of a simple function 80

The L1-seminorm 82

The Bochner-Lebesgue integral 84

The completeness of L1 87

Elementary properties of integrals 88

Convergence in L1 91

3 Convergence theorems 97

Integration of nonnegative ？-valued functions 97

The monotone convergence theorem 100

Fatou's lemma 101

Integration of ？-valued functions 103

Lebesgue's dominated convergence theorem 104

Parametrized integrals 107

4 Lebesgue spaces 110

Essentially bounded functions 110

The H？lder and Minkowski inequalities 111

Lebesgue spaces are complete 114

Lp-spaces 116

Continuous functions with compact support 118

Embeddings 119

Continuous linear functionals on Lp 121

5 The n-dimensional Bochner-Lebesgue integral 128

Lebesgue measure spaces 128

The Lebesgue integral of absolutely integrable functions 129

A characterization of Riemann integrable functions 132

6 Fubini's theorem 137

Maps defined almost everywhere 137

Cavalieri's principle 138

Applications of Cavalieri's principle 141

Tonelli's theorem 144

Fubini's theorem for scalar functions 145

Fubini's theorem for vector-valued functions 148

Minkowski's inequality for integrals 152

A characterization of Lp(Rm+n,E) 157

A trace theorem 158

7 The convolution 162

Defining the convolution 162

The translation group 165

Elementary properties of the convolution 168

Approximations to the identity 170

Test functions 172

Smooth partitions of unity 173

Convolutions of E-valued functions 177

Distributions 177

Linear differential operators 181

Weak derivatives 184

8 The substitution rule 191

Pulling back the Lebesgue measure 191

The substitution rule:general case 195

Plane polar coordinates 197

Polar coordinates in higher dimensions 198

Integration of rotationally symmetric functions 202

The substitution rule for vector-valued functions 203

9 The Fourier transform 206

Definition and elementary properties 206

The space of rapidly decreasing functions 208

The convolution algebra S 211

Calculations with the Fourier transform 212

The Fourier integral theorem 215

Convolutions and the Fourier transform 218

Fourier multiplication operators 220

Plancherel's theorem 223

Symmetric operators 225

The Heisenberg uncertainty relation 227

Chapter Ⅺ Manifolds and differential forms 235

1 Submanifolds 235

Definitions and elementary properties 235

Submersions 241

Submanifolds with boundary 246

Local charts 250

Tangents and normals 251

The regular value theorem 252

One-dimensional manifolds 256

Partitions of unity 256

2 Multilinear algebra 260

Exterior products 260

Pull backs 267

The volume element 268

The Riesz isomorphism 271

The Hodge star operator 273

Indefinite inner products 277

Tensors 281

3 The local theory of differential forms 285

Definitions and basis representations 285

Pull backs 289

The exterior derivative 292

The Poincaré lemma 295

Tensors 299

4 Vector fields and differential forms 304

Vector fields 304

Local basis representation 306

Differential forms 308

Local representations 311

Coordinate transformations 316

The exterior derivative 319

Closed and exact forms 321

Contractions 322

Orientability 324

Tensor fields 330

5 Riemannian metrics 333

The volume element 333

Riemannian manifolds 337

The Hodge star 348

The codifferential 350

6 Vector analysis 358

The Riesz isomorphism 358

The gradient 361

The divergence 363

The Laplace-Beltrami operator 367

The curl 372

The Lie derivative 375

The Hodge-Laplace operator 379

The vector product and the curl 382

Chapter Ⅻ Integration on manifolds 391

1 Volume measure 391

The Lebesgue σ-algebra of M 391

The definition of the volume measure 392

Properties 397

Integrability 398

Calculation of several volumes 401

2 Integration of differential forms 407

Integrals of m-forms 407

Restrictions to submanifolds 409

The transformation theorem 414

Fubini's theorem 415

Calculations of several integrals 418

Flows of vector fields 421

The transport theorem 425

3 Stokes's theorem 430

Stokes's theorem for smooth manifolds 430

Manifolds with singularities 432

Stokes's theorem with singularities 436

Planar domains 439

Higher-dimensional problems 441

Homotopy invariance and applications 443

Gauss's law 446

Green's formula 448

The classical Stokes's theorem 450

The star operator and the coderivative 451

References 457