On the range of a vector measure

Abstract

Let ( Ω , Σ , μ ) (\Omega ,\Sigma ,\mu ) be a finite measure space, let Z Z be a Banach space, and let ν : Σ → Z ∗ \nu :\Sigma \to Z^* be a countably additive μ \mu -continuous vector measure. Let X ⊆ Z ∗ X \subseteq Z^* be a norm-closed subspace which is norming for Z Z . Write σ ( Z , X ) \sigma (Z,X) (resp., μ ( X , Z ) \mu (X,Z) ) to denote the weak (resp., Mackey) topology on Z Z (resp., X X ) associated to the dual pair ⟨ X , Z ⟩ \langle X,Z\rangle . Suppose that, either ( Z , σ ( Z , X ) ) (Z,\sigma (Z,X)) has the Mazur property, or ( B X ∗ , w ∗ ) (B_{X^*},w^*) is convex block compact and ( X , μ ( X , Z ) ) (X,\mu (X,Z)) is complete. We prove that the range of ν \nu is contained in X X if, for each A ∈ Σ A\in \Sigma with μ ( A ) > 0 \mu (A)>0 , the w ∗ w^* -closed convex hull of { ν ( B ) μ ( B ) : B ∈ Σ , B ⊆ A , μ ( B ) > 0 } \{\frac {\nu (B)}{\mu (B)}: \, B\in \Sigma , \, B \subseteq A, \, \mu (B)>0\} intersects X X . This extends results obtained by Freniche [Proc. Amer. Math. Soc. 107 (1989), no. 1, pp. 119–124] when Z = X ∗ Z=X^* .