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

Abstract

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].