Two generalizations of Komlós' theorem with lower closure-type applications

Two generalizations of Komlos' theorem for functions and multifunctions with values in a Banach space are presented. The first generalization is partly new and the second one is a Komlos-type result of a completely new nature. It requires the Radon-Nikodym property for the Banach space and its dual. In both cases our approach relies on using Komlos' theorem by means of a diagonal extraction argument, as introduced in [5, 6, 7]. Two quite general lower closure-type results, which follow immediately from our main results, are shown to generalize or substantially extend a number of results in the literature