A Grothendieck-type theorem for the space of totally measurable functions

Abstract

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X * stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X) * and B(Σ, X) ** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X *) denote the Banach space of all countably additive vector measures ν: Σ → X * of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X *), B(Σ, X))-sequential compactness in bvca(Σ, X *) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ( B(Σ, X) *, B(Σ, X))-sequentially compact subset of B(Σ, X) c ~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ( B(Σ, X) *, B(Σ, X) **)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ( B(Σ, X) *, B(Σ, X))-convergent sequences in B(Σ, X) c ~ are σ( B(Σ, X) *, B(Σ, X) **)-convergent.