Applications of the Theory of Orlicz Spaces to Vector Measures

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 {d(e′om)dλ:m∈M,e′∈D}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\frac{{d\left( {e'om} \right)}}{{d\lambda }}:m \in M,e' \in D} \right\}$$\end{document} 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.