Vector measure Banach spaces containing a complemented copy of 𝑐₀

Let X X a Banach space and Σ \Sigma a σ \sigma -algebra of subsets of a set Ω \Omega . We say that a vector measure Banach space ( M ( Σ , X ) , ‖ ⋅ ‖ M ) (\mathcal {M} (\Sigma , X ) , \Vert \cdot \Vert _\mathcal {M }) has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure m : Σ ⟶ X m : \Sigma \longrightarrow X , for which there exists a bounded sequence ( m n ) (m_{n}) in M ( Σ , X ) \mathcal {M } (\Sigma , X ) verifying lim n → ∞ m n ( A ) = m ( A ) \displaystyle \lim _{n \to \infty } m_{n} ( A ) = m(A) for all A ∈ Σ A \in \Sigma , must belong to M ( Σ , X ) \mathcal {M} (\Sigma , X) . Among other results, we prove that, if M ( Σ , X ) \mathcal {M}(\Sigma , X) is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of c 0 c_{0} , then X X contains a copy of c 0 c_{0} .