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.
@article{1176992264,
author = {Talagrand, Michel},
title = {Measurability Problems for Empirical Processes},
journal = {Ann. Probab.},
volume = {15},
number = {4},
year = {1987},
pages = { 204-212},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992264}
}
Talagrand, Michel. Measurability Problems for Empirical Processes. Ann. Probab., Tome 15 (1987) no. 4, pp. 204-212. http://gdmltest.u-ga.fr/item/1176992264/