Measure and integration in bornological spaces. (Spanish: Medida e integración en espacios bornológicos).

Abstract

Let E be a vector space with a convex "bornology'', in the sense of H. Hogbe-Nlend [Théorie des bornologies et applications, Lecture Notes in Math., 213, Springer, Berlin, 1971. If Ω is a set and Σ a σ -algebra of P(Ω) , a map m:Σ→E such that m(∅)=0 is called a bornological measure if, for any sequence A n ∈Σ of pairwise disjoint sets, one has (⋃ ∞ 1 A n )=∑ ∞ 1 m(A n ) for the Mackey convergence. The aim of the paper is to give conditions for the existence of a bounded absolutely convex set B such that, if m(Σ)⊂R ⋅ B , then m is a classical vector measure with values in E B (=R ⋅ B normed with the gauge of B ); for instance, if E B is a Banach space: (a) l ∞ ⊄E B or (b) F closed and separable in E B implies that F∩B is closed for σ(E,E × ) , where E × is the set of bounded elements of E ∗ . Then the author gives a notion of a measurable function f with respect to a bornology as above, and gives sufficient conditions to have f Bochner measurable with respect to some E B as above.