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Resumen:
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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 ? is a set and ? a ? -algebra of P(?) , a map m:??E such that m(?)=0 is called a bornological measure if, for any sequence A n ?? 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(?)?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 ?(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.
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