Abstract

For any p e [1, oo], we prove that the set of simple functions taking at most k different values is proximinal in Lp for all k > 1. Moreover, if 1 < p < oo, we prove that these sets are approximatively norm-compact. We introduce the class of uniformly approximable subsets of Lp, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the p-variation if p e [1, oo) and in terms of covering numbers if p = oo. We study properties of uniformly approximable sets. In particular, we prove that the convex hull of a uniformly approximable bounded set is also uniformly approximable and that this class is stable under Holder transformations.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).