Two Generalizations of Komlós' Theorem with Lower Closure-Type Applications

Two generalizations of Komlós’ 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 Komlós-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 Komlós’ 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.