On vector measures with values in 𝑐₀(𝜅)

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Let ν \nu be a vector measure defined on a σ \sigma -algebra Σ \Sigma and taking values in a Banach space. We prove that if ν \nu is homogeneous and L 1 ( ν ) L_1(\nu ) is non-separable, then there is a vector measure ν ~ : Σ → c 0 ( κ ) \tilde {\nu }:\Sigma \to c_0(\kappa ) such that L 1 ( ν ) = L 1 ( ν ~ ) L_1(\nu )=L_1(\tilde {\nu }) with equal norms, where κ \kappa is the density character of L 1 ( ν ) L_1(\nu ) . This is a non-separable version of a result of G.P. Curbera [Pacific J. Math. 162 (1994), pp. 287–303].