Vector measure Banach spaces containing a complemented copy of $c_{0}$

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