Applications of the Theory of Orlicz Spaces to Vector Measures

Abstract

Let (Ω, Σ, λ) be a finite complete measure space, (E, ξ) be a sequentially complete locally convex Hausdorff space and E′ be its topological dual. Let caλ (Σ, E) stand for the space of all λ-absolutely continuous measures m: Σ → E. We show that a uniformly bounded subset M of caλ (Σ, E) is uniformly λ-absolutely continuous if and only if for every equicontinuous subset D of E′, there exists a submultiplicative Young function φ such that the set $$\\left\\{ {\\frac{{d\\left( {e om} \\right)}}{{d\\lambda }}:m \\in M,e \\in D} \\right\\}$${d(e′om)dλ:m∈M,e′∈D} is relatively weakly compact in the Orlicz space Lφ(λ). As a consequence, we present a generalized Vitali–Hahn–Saks theorem on the setwise limit of a sequence of λ-absolutely continuous vector measures in terms of Orlicz spaces.