On Vector Measures, Uniform Integrability and Orlicz Spaces

Given a Banach space X and a probability space (Omega, Sigma, mu), we characterize the countable additivity of the Dunford integral for Dunford integrable functions taking values in X as those weakly measurable functions f : Omega -> X for which {x* f : x* is an element of B-X*} is relatively weakly compact in some separable Orlicz space L-phi(mu). 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 {x f : is an element of B-X}, whenever f : Omega -> X* is Gel'fand integrable.