Integration

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 Fr chet spaces, as well as "e;weak"e; spaces and distribution spaces. We integrate "e;integrable measures"e;, which are equivalent to "e;classes of integrable functions which are a.e. equals"e; when E is a Fr chet 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 Fr chet space, we study the equivalence of the two definitions and the properties related to dominated convergence.