Jacques Simon, CNRS, France.
ISBN : 9781786300133
Publication Date : February 2026
Hardcover 440 pp
165.00 USD
Co-publisher
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(?;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(?;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.
Part 1. Integration.
1. Integration of Continuous Functions.
2. Measurable Sets.
3. Measures.
4. Integrable Measures.
5. Integration of Integrable Measures.
6. Properties of the Integral.
7. Change of Variables.
8. Multivariable Integration.
Part 2. Lebesgue Spaces.
9. Inequalities.
10. Lp(!; E) Spaces.
11. Dependence on pand !, Local Spaces.
12. Image Under a Linear Mapping.
13. Various Operations.
14. Change of Variable, Weightings.
15. Compact Sets.
16. Duals.
Part 3. Integrable Function.
17. Measurable Functions.
18. Applications.
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.
