Let $(\Omega,\Sigma,\mu)$ be a complete finite 
measure space, $X$ a separable Banach space and 
$Z$ a proper closed linear subspace of $X^{\ast}$. 
If the subspace of $L_{w^{\ast}}^{\infty}(\mu,X^{\ast})$ 
(the Banach space of all [classes of] essentially bounded 
$X^{\ast}$-valued weak* measurable functions defined on 
$\Omega$ equipped with its usual norm) consisting of all 
those $Z$-valued functions contains a complemented copy of 
$c_{0}$, we show in this note that $Z$ contains a copy of 
$c_{0}$.