A unified approach to several results involving integrals of multifunctions

Abstract

A well-known equivalence of randomization result of Wald and Wolfowitz states that any Young measure can be regarded as a probability measure on the set of all measurable functions. Here we give a sucient condition for the Young measure to be equivalent to a probability measure on the set of all integrable selectors of a given multifunction. In this way, Aumann's identity for integrals of multifunctions can be interpreted in a novel fashion. By additionally applying a fundamental result from Young measure theory to uniformly L1-bounded sequences of functions, Fatou's lemma in several dimensions, which is formulated in terms of the integral of a Kuratowski limes superior multifunction, can be proven in a new fashion. Also, a natural extension of these arguments leads to a generalization of a recent result by Artstein and Rzezuchowski [3].