This book presents a simple and novel theory of integration, both real and vectorial, particularly suitable for the study of PDEs. This theory allows for integration with values in a Neumann space E, i.e. in which all Cauchy sequences converge, encompassing Neumann and Fréchet spaces, as well as "weak" spaces and distribution spaces.
We integrate "integrable measures", which are equivalent to "classes of integrable functions which are a.e. equals" when E is a Fréchet space. More precisely, we associate the measure f with a class f, where f(u) is the integral of fu for any test function u. The classic space Lp(Ω;E) is the set of f, and ours is the set of f; these two spaces are isomorphic.
Integration studies, in detail, for any Neumann space E, the properties of the integral and of Lp(Ω;E): regularization, image by a linear or multilinear application, change of variable, separation of multiple variables, compacts and duals. When E is a Fréchet space, we study the equivalence of the two definitions and the properties related to dominated convergence.
Table of Contents:
Introduction xi
List of Notations and Figures xv
Part 1. Integration 1
Chapter 1. Integration of Continuous Functions 3
1.1. Neumann spaces 3
1.2. Continuous mappings 7
1.3. Cauchy integral of a uniformly continuous function 10
1.4. Some properties of the integral 13
1.5. Dependence of the integral on the domain of integration 16
1.6. Continuity of the integral 19
1.7. Successive integration 21
Chapter 2. Measurable Sets 23
2.1. Why introduce measurable sets? 23
2.2. Some properties of the measure of an open set 25
2.3. Definition of measurable sets and their measure 28
2.4. First properties of the measure 32
2.5. Additivity of the measure 34
2.6. Countable union and countable intersection of measurable sets 37
2.7. Continuity of the measure 40
2.8. Translation invariance and the product measure 44
2.9. Negligible sets 47
Chapter 3. Measures 51
3.1. Space of measuresM(Ω;E) 51
3.2. Equicontinuity of bounded subsets ofM(Ω;E) 54
3.3. Sequential completeness ofM(Ω;E) 57
3.4. Continuity of the ⟨ , ⟩ mapping 59
3.5. Identification of continuous functions with measures 60
3.6. Regularization of measures 66
3.7. Regularization of functions 73
Chapter 4. Integrable Measures 79
4.1. Definition of integrable measures 79
4.2. Space of integrable measures L1(Ω;E) 82
4.3. Some properties of L1(Ω;E) 85
4.4. Regularization in L1(Ω;E) 87
4.5. Sequential completeness of L1(Ω;E) 88
Chapter 5. Integration of Integrable Measures 91
5.1. Integral of an integrable measure 91
5.2. Linearity and continuity of the integral 95
5.3. Positive measures, real-valued integrals 97
5.4. Examples of value spaces 100
5.5. The case where E is not a Neumann space 101
Chapter 6. Properties of the Integral 105
6.1. Additivity with respect to the domain of integration 105
6.2. Continuity with respect to the domain of integration 109
6.3. Contribution of negligible sets 113
6.4. Image of a measure under a linear mapping 114
6.5. Image under a linear mapping 116
6.6. Restriction and support 119
6.7. Differentiation under the integral sign 121
Chapter 7. Change of Variables 123
7.1. Image of a measurable set 123
7.2. Determinant of d vectors 125
7.3. Measure of a parallelepiped 127
7.4. Change of variable in the Cauchy integral 130
7.5. Change of variable in a measure 137
7.6. Change of variable in an integrable measure 141
7.7. Product of a measure with a continuous function 143
7.8. Change of variable in an integral 146
7.9. Affine change of variables 148
Chapter 8. Multivariable Integration 151
8.1. Permutation of variables in a measure of measures 151
8.2. Integration of an integrable measure of measures 152
8.3. Separation of variables in an integral of a continuous function 155
8.4. Separation of variables of a measure 158
8.5. Separation of variables 161
8.6. Fubini’s theorem 164
Part 2. Lebesgue Spaces 169
Chapter 9. Inequalities 171
9.1. Elementary inequalities 171
9.2. Inequalities for continuous functions 174
9.3. Young’s convolution inequality 177
9.4. Properties of regularizations of continuous functions 179
Chapter 10. Lp(Ω;E) Spaces 183
10.1. Definition of Lp(Ω;E) 183
10.2. Separability of Lp(Ω;E) 188
10.3. Some properties of Lp(Ω;E) 189
10.4. Properties of L∞(Ω;E) 192
10.5. Approximation via regularizations and density 196
10.6. Completeness of Lp(Ω;E) 199
10.7. Remarks on methods of construction 203
Chapter 11. Dependence on p and Ω, Local Spaces 207
11.1. Dependence on p 207
11.2. Lp loc(Ω;E) spaces 211
11.3. Localization–extension 217
11.4. Dependence on Ω 220
11.5. Infinite gluing on Ω and continuity in p 223
Chapter 12. Image Under a Linear Mapping 229
12.1. Image under a linear mapping and dependence on E 229
12.2. Image under a multilinear mapping 233
12.3. Images in Banach and Hilbert spaces 239
12.4. Images in local spaces 242
Chapter 13. Various Operations 245
13.1. Image under a semi-norm of E 245
13.2. Powers 249
13.3. Extensions 252
13.4. Step measures 254
13.5. Density and separability 258
13.6. Limit of a bounded sequence in L∞(Ω;E) 261
Chapter 14. Change of Variable, Weightings 263
14.1. Change of variable 263
14.2. Regrouping and separation of variables 266
14.3. Permutation of variables 273
14.4. Weightings of measures 275
14.5. Weightings 278
Chapter 15. Compact Sets 283
15.1. Preliminaries 283
15.2. Compact subsets of Lp(Ω;E) 286
15.3. Special cases of compactness 290
15.4. Compact subsets of Lp loc(Ω;E) 295
15.5. Compactness in intermediate spaces 297
Chapter 16. Duals 301
16.1. Uniform convexity of Lp(Ω;E) 301
16.2. Canonical injection from Lp' (Ω;E') into the dual of Lp(Ω;E) 310
16.3. Riesz representation theorems 315
16.4. Riesz–Fréchet theorem 320
16.5. Weak topology of Lp(Ω;E) 322
16.6. ∗Weak topology of L∞(Ω;E) 324
Part 3. Integrable Functions 329
Chapter 17. Measurable Functions 331
17.1. Measurable functions 331
17.2. Integral of a positive measurable function 337
17.3. Dominated convergence of positive functions 342
17.4. Spaces of classes of integrable functions 347
17.5. Completion and approximation in spaces of classes of functions 351
17.6. Some properties of spaces of classes of functions 357
17.7. Lebesgue points 359
17.8. Measures associated to classes of functions 363
17.9. Identity of the spaces of measures 366
Chapter 18. Applications 371
18.1. Equi-integrability 371
18.2. Dominated convergence 374
18.3. Image under a continuous mapping 377
18.4. Continuity with respect to increasing p (again) 380
18.5. Riesz representation theorem (again) 383
Appendix. Reminders 391
Bibliography 405
Index 409
