A course on integration theory : including more than 150 exercises with detailed answers

Preface . . xi

1 General Theory of Integration
1.1 Measurable spaces, o-algebras . . 1
1.2 Measurable spaces and topological spaces . . 3
1.3 Structure of measurable functions . . 15
1.4 Positive measures . . 17
1.5 Integrating non-negative functions . . 22
1.6 Three basic convergence theorems . . 28
1.7 Space L1(u) and negligible sets . . 34
1.8 Notes . . 39
1.9 Exercises . . 41

2 Actual Construction of Measure Spaces
2.1 Partitions of unity . . 67
2.2 The Riesz Markov representation theorem . .70
2.3 Producing positive Radon measures . . 82
2.4 The Lebesgue measure on Rm, properties arid characterization . . 86
2.5 Caratheodory theorem on outer measures . . 93
2.6 Hausdorff measures, Hausdorff dimension . . 96
2.7 Notes . . 102
2.8 Exercises . . 103

3 Spaces of Integrable Functions
3.1 Convexity inequalities (Jensen, Holder, Minkowski) . . 125
3.2 Lp spaces . . 132
3.3 Integrals depending on a parameter . . 140
3.4 Continuous functions in Lp spaces . . 147
3.5 On various notions of convergence . . 152
3.6 Notes . . 154
3.7 Exercises . . 155

4 Integration on a Product Space
4.1 Product of measurable spaces . . 189
4.2 Tensor product of sigma-finite measures . . 192
4.3 The Lebosgue measure 011 Rm and tensor products . . 199
4.4 Notes . . 200
4.5 Exercises . . 201

5 Diffeomorphisms of Open Subsets of Rn and Integration
5.1 Differentiability . . 219
5.2 Linear transformations . . 223
5.3 Change-of-vanables formula . . 228
5.4 Examples, polar coordinates in Rn . . 233
5.5 Integration on a C1 hypersurface of the Euclidean Rn . . 238
5.6 More on Hausdorff measures on Rn . . 242
5.7 Cantor sets . . 249
5.8 Category arid measure . . 262
5.9 Notes . . 264
5.10 Exercises . . 266

6 Convolution
6.1 The Banach algebra L1(Rn) . . 283
6.2 Lp Estimates for convolution, Young's inequality . . 288
6.3 Weak Lp spaces . . 293
6.4 The Hardy Littlewood-Sobolev inequality . . 297
6.5 Notes . . 301
6.6 Exercises . . 302

7 Complex Measures
7.1 Complex measures . . 317
7.2 Total variation of a complex measure . . 319
7.3 Absolute continuity, mutually singular measures . . 321
7.4 Radon Nikodym theorem . . 323
7.5 The dual of LP(X, M,u), 1 < p < +00 . .  327
7.6 Notes . . 333
7.7 Exercises . . 334

8 Basic Harmonic Analysis on Rn
8.1 Fouier transform of tempered distributions . . 343
8.2 The Poisson summation formula . . 357
8.3 Periodic distributions . . 361
8.4 Notes . . 365
8.5 Exercises . . 366

9 Classical Inequalities
9.1 Riesz-Thorin interpolation theorem . . 371
9.2 Marcinkiewicz Interpolation Theorem . . 380
9.3 Maximal function . . 383
9.4 Lebesgue differentiation theorem, Lebesgue points . . 386
9.5 Gagliardo-Nirenberg inequality . . . . 389
9.6 Sobolev spaces, Sobolev injection theorems . . 394
9.7 Notes . . 399
9.8 Exorcises . . 400

10 Appendix
10.1 Set theory, cardinals, ordinals . . 407
10.2 Topological matters (Tychonoff, Hahn-Bariach, Baire) . . 425
10.3 Duality in Banach spaces (weak convergence, reflexivity) . . 440
10.4 Calculating antiderivatives (classics, Abelian, Gaussian) . . 448
10.5 Some special functions (logarithm, Gamma function, Laplacean) . . 461
10.6 Classical volumes and areas (balls, spheres, cones, polyhedra) . . 474

Bibliography . . 481
Index . . 487