Universidad de Huelva.

Let X a Banach space and Σ a σ-algebra of subsets of a set Ω. We say that a vector measure Banach space (M(Σ, X), ‖ · ‖M) has the bounded Vitaly-Hahn-Sacks Property if satisfies the following condition: Every vector measure m: Σ − → X, for which there exists a bounded sequence (mn) in M(Σ, X) verifying lim n→∞ mn(A) = m(A) for all A ∈ Σ, must belong to M(Σ, X). Among other results, we prove that, if M(Σ, X) is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of c0, then X contains a copy of c0.