On Pettis integrals with separable range
Several techniques have been developed to study Pettis integrability of weakly measurable functions with values in Banach spaces. As shown by M. Talagrand [Ta], it is fruitful to regard a weakly measurable mapping as a pointwise compact set of measurable functions - its Pettis integrability is then a purely measure-theoretic question of an appropriate continuity of a measure. On the other hand, properties of weakly measurable functions can be translated into the language of topological measure theory by means of weak Baire measures on Banach spaces. This approach, originated by G. A. Edgar [E1, E2], was remarkably developed by M. Talagrand. Following this idea, we show that the Pettis Integral Property of a Banach space E, together with the requirement of separability of E-valued Pettis integrals, is equivalent to the fact that every weak Baire measure on E is, in a certain weak sense, concentrated on a separable subspace. We base on a lemma which is a version of Talagrand's Lemma 5-1-2 from [Ta]. Our lemma easily yields a sequential completeness of the spaces of Grothendieck measures, a related result proved by Pallarés-Vera [PV]. We also present two results on Pettis integrability in the spaces of continuous functions.