On Vector Measures, Uniform Integrability and Orlicz Spaces

Abstract

Given a Banach space X and a probability space (Ω, Σ, μ), we characterize the countable additivity of the Dunford integral for Dunford integrable functions taking values in X as those weakly measurable functions f:Ω→X for which {x*f:x*∈B _X*} is relatively weakly compact in some separable Orlicz space L ^ϕ(μ). We also provide a characterization of the Pettis integral of Dunford integrable functions by mean of weak compactness in separable Orlicz spaces and give a necessary and sufficient condition for the uniform integrability of {xf:x∈B _x }, whenever f:Ω→X* is Gel’fand integrable.