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