The Glivenko-Cantelli Problem

Abstract

We give a new type of characterization of the Glivenko-Cantelli classes. In the case of a class $\mathscr{L}$ of sets, the characterization is closely related to the configuration that the sets of $\mathscr{L}$ can have. It allows one to decide simply whether a given class is a Glivenko-Cantelli class. The characterization is based on a new measure theoretic analysis of sets of measurable functions. This analysis also gives an approximation theorem for Glivenko-Cantelli classes, sharpenings of the Vapnik-Cervonenkis criteria and the value of the asymptotic discrepancy for classes that are not Glivenko-Cantelli. An application is given to the law of large numbers in a Banach space for functions that need not be random variables.