Derivability, variation and range of a vector measure
We prove that the range of a vector measure determines the σ-finiteness of its variation and the derivability of the measure. Let F and G be two countably additive measures with values in a Banach space such that the closed convex hull of the range of F is a translate of the closed convex hull of the range of G; then F has a σ-finite variation if and only if G does, and F has a Bochner derivative with respect to its variation if and only if G does. This complements a result of [Ro] where we proved that the range of a measure determines its total variation. We also give a new proof of this fact.