Measurability Problems for Empirical Processes

Abstract

To a class $\mathscr{F}$ of bounded functions on a probability space we associate two classes $\mathscr{F}_r$ and $\mathscr{F}_s$. The class $\mathscr{F}$ is a Donsker class if and only if $\mathscr{F}_r$ and $\mathscr{F}_s$ are Donsker classes. The class $\mathscr{F}_r$ corresponds to a separable version of the empirical process. It is obtained by applying a special type of lifting to $\mathscr{F}$. The class $\mathscr{F}_s$ consists of positive functions that are zero almost surely. It concentrates the pathology of $\mathscr{F}$ with respect to measurability. We use this method to prove without any measurability assumption a general contraction principle for processes that satisfy the central limit theorem.