On the range of a vector measure

Abstract

Let (Ω, Σ, µ) be a finite measure space, Z be a Banach space and ν : Σ → Z * be a countably additive µ-continuous vector measure. Let X ⊆ Z * be a norm-closed subspace which is norming for Z. Write σ(Z, X) (resp. µ(X, Z)) to denote the weak (resp. Mackey) topology on Z (resp. X) associated to the dual pair X, Z . Suppose that, either (Z, σ(Z, X)) has the Mazur property, or (B X * , w * ) is convex block compact and (X, µ(X, Z)) is complete. We prove that the range of ν is contained in X if, for each A ∈ Σ with µ(A) > 0, the w * -closed convex hull of { ν(B) µ(B) : B ∈ Σ, B ⊆ A, µ(B) > 0} intersects X. This extends results obtained by Freniche [Proc. Amer. Math. Soc. 107 (1989), no. 1, 119-124] when Z = X * . 2010 Mathematics Subject Classification. 46A50, 46G10.