Integration

Jacques Simon is Director Emeritus of Research at the CNRS, France. His research focuses on partial differential equations, particularly on the spaces used by these equations and on shape optimization.

• Introduction xiList of Notations and Figures xvPart 1. Integration 1Chapter 1. Integration of Continuous Functions 31.1. Neumann spaces 31.2. Continuous mappings 71.3. Cauchy integral of a uniformly continuous function 101.4. Some properties of the integral 131.5. Dependence of the integral on the domain of integration 161.6. Continuity of the integral 191.7. Successive integration 21Chapter 2. Measurable Sets 232.1. Why introduce measurable sets? 232.2. Some properties of the measure of an open set 252.3. Definition of measurable sets and their measure 282.4. First properties of the measure 322.5. Additivity of the measure 342.6. Countable union and countable intersection of measurable sets 372.7. Continuity of the measure 402.8. Translation invariance and the product measure 442.9. Negligible sets 47Chapter 3. Measures 513.1. Space of measuresM(Ω;E) 513.2. Equicontinuity of bounded subsets ofM(Ω;E) 543.3. Sequential completeness ofM(Ω;E) 573.4. Continuity of the ⟨ , ⟩ mapping 593.5. Identification of continuous functions with measures 603.6. Regularization of measures 663.7. Regularization of functions 73Chapter 4. Integrable Measures 794.1. Definition of integrable measures 794.2. Space of integrable measures L1(Ω;E) 824.3. Some properties of L1(Ω;E) 854.4. Regularization in L1(Ω;E) 874.5. Sequential completeness of L1(Ω;E) 88Chapter 5. Integration of Integrable Measures 915.1. Integral of an integrable measure 915.2. Linearity and continuity of the integral 955.3. Positive measures, real-valued integrals 975.4. Examples of value spaces 1005.5. The case where E is not a Neumann space 101Chapter 6. Properties of the Integral 1056.1. Additivity with respect to the domain of integration 1056.2. Continuity with respect to the domain of integration 1096.3. Contribution of negligible sets 1136.4. Image of a measure under a linear mapping 1146.5. Image under a linear mapping 1166.6. Restriction and support 1196.7. Differentiation under the integral sign 121Chapter 7. Change of Variables 1237.1. Image of a measurable set 1237.2. Determinant of d vectors 1257.3. Measure of a parallelepiped 1277.4. Change of variable in the Cauchy integral 1307.5. Change of variable in a measure 1377.6. Change of variable in an integrable measure 1417.7. Product of a measure with a continuous function 1437.8. Change of variable in an integral 1467.9. Affine change of variables 148Chapter 8. Multivariable Integration 1518.1. Permutation of variables in a measure of measures 1518.2. Integration of an integrable measure of measures 1528.3. Separation of variables in an integral of a continuous function 1558.4. Separation of variables of a measure 1588.5. Separation of variables 1618.6. Fubini’s theorem 164Part 2. Lebesgue Spaces 169Chapter 9. Inequalities 1719.1. Elementary inequalities 1719.2. Inequalities for continuous functions 1749.3. Young’s convolution inequality 1779.4. Properties of regularizations of continuous functions 179Chapter 10. Lp(Ω;E) Spaces 18310.1. Definition of Lp(Ω;E) 18310.2. Separability of Lp(Ω;E) 18810.3. Some properties of Lp(Ω;E) 18910.4. Properties of L∞(Ω;E) 19210.5. Approximation via regularizations and density 19610.6. Completeness of Lp(Ω;E) 19910.7. Remarks on methods of construction 203Chapter 11. Dependence on p and Ω, Local Spaces 20711.1. Dependence on p 20711.2. Lp loc(Ω;E) spaces 21111.3. Localization–extension 21711.4. Dependence on Ω 22011.5. Infinite gluing on Ω and continuity in p 223Chapter 12. Image Under a Linear Mapping 22912.1. Image under a linear mapping and dependence on E 22912.2. Image under a multilinear mapping 23312.3. Images in Banach and Hilbert spaces 23912.4. Images in local spaces 242Chapter 13. Various Operations 24513.1. Image under a semi-norm of E 24513.2. Powers 24913.3. Extensions 25213.4. Step measures 25413.5. Density and separability 25813.6. Limit of a bounded sequence in L∞(Ω;E) 261Chapter 14. Change of Variable, Weightings 26314.1. Change of variable 26314.2. Regrouping and separation of variables 26614.3. Permutation of variables 27314.4. Weightings of measures 27514.5. Weightings 278Chapter 15. Compact Sets 28315.1. Preliminaries 28315.2. Compact subsets of Lp(Ω;E) 28615.3. Special cases of compactness 29015.4. Compact subsets of Lp loc(Ω;E) 29515.5. Compactness in intermediate spaces 297Chapter 16. Duals 30116.1. Uniform convexity of Lp(Ω;E) 30116.2. Canonical injection from Lp' (Ω;E') into the dual of Lp(Ω;E) 31016.3. Riesz representation theorems 31516.4. Riesz–Fréchet theorem 32016.5. Weak topology of Lp(Ω;E) 32216.6. ∗Weak topology of L∞(Ω;E) 324Part 3. Integrable Functions 329Chapter 17. Measurable Functions 33117.1. Measurable functions 33117.2. Integral of a positive measurable function 33717.3. Dominated convergence of positive functions 34217.4. Spaces of classes of integrable functions 34717.5. Completion and approximation in spaces of classes of functions 35117.6. Some properties of spaces of classes of functions 35717.7. Lebesgue points 35917.8. Measures associated to classes of functions 36317.9. Identity of the spaces of measures 366Chapter 18. Applications 37118.1. Equi-integrability 37118.2. Dominated convergence 37418.3. Image under a continuous mapping 37718.4. Continuity with respect to increasing p (again) 38018.5. Riesz representation theorem (again) 383Appendix. Reminders 391Bibliography 405Index 409