Convergence of adapted sequences of Pettisintegrable functions.

Abstract

When considering adapted sequences of Pettis-integrable functions with values in a Banach space we are dealing with the following problem: when do we have a strongly measurable Pettis-integrable limit? Here the limit can be taken in the strong or weak sense a.e. or in the sense of the Pettis-topology. Not many results in this area are known so far. In this paper we give some pointwise convergence results of martingales, amarts, weak sequential amarts and pramarts consisting of strongly measurable Pettis-integrable functions. Also the Pettis convergence result of Musial for amarts is extended. The results are preceded by a preliminary study of some vector measure notions such as Pettis uniform integrability and σ-bounded variation. We give a new proof of the result of Thomas stating that in every infinite dimensional Banach space one can find a vector measure which is not of σ-bounded variation.