On vector measures with values in c₀(κ)

Let ν be a vector measure defined on a σ-algebra Σ and taking values in a Banach space. We prove that if ν is homogeneous and L₁(ν) is non-separable, then there is a vector measure ν̃:Σ → c₀(κ) such that L₁(ν)=L₁(ν̃) with equal norms, where κ is the density character of L₁(ν). This is a non-separable version of a result of [G.P. Curbera, Pacific J. Math. 162 (1994), no. 2, 287--303].