On vector measures with values in $c_0(\kappa)$

Let $\nu$ be a vector measure defined on a $\sigma$-algebra $\Sigma$ and taking values in a Banach space. We prove that if $\nu$ is homogeneous and $L_1(\nu)$ is non-separable, then there is a vector measure $\tilde{\nu}:\Sigma \to c_0(\kappa)$ such that $L_1(\nu)=L_1(\tilde{\nu})$ with equal norms, where $\kappa$ is the density character of $L_1(\nu)$. This is a non-separable version of a result of [G.P. Curbera, Pacific J. Math. 162 (1994), no. 2, 287--303].