Overview

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.

Full Product Details

Author:   Jacques Simon (CNRS, France)
Publisher:   ISTE Ltd and John Wiley & Sons Inc
Imprint:   ISTE Ltd and John Wiley & Sons Inc
Weight:   0.666kg
ISBN:  

9781786300133

ISBN 10:   1786300133
Pages:   448
Publication Date:   19 January 2026
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available.

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

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Author Information

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.

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