Equally Distributed Correspondences

The study of measurable and integrable selections from a correspondence (set-valued function) has been of interest for some time [e.g., see Aumann (1965), Hildenbrand (1974), etc.]. Here we take up the study of distributions of such selections. The motivation for doing this comes from Mathematical Economics [see Kannai (1970, section 7); the reader interested in the economic applications is referred to Hart et al. (1974)]. The first question we consider is the following: Does the distribution of a correspondence determine the distributions of its selections ? If the underlying space contains atoms, then of course we cannot expect an affirmative answer (e.g., consider the correspondence whose image is always the set {1,2}, once on a space consisting of one atom, and once on a space consisting of two atoms). But what if the underlying space is atomless ? Consider the following example [which is essentially a reformulation of an example due to G. Debreu see Kannai (1970, section 7)]. Let VI(t) = {t, -t> for all t E [O, 11,