Proximinality and uniformly approximable sets in Lp

Abstract

For any $p\in [1,\infty ]$, we prove that the set of simple functions taking at most $k$ different values is proximinal in $L^p$ for all $k\ge 1$. Moreover, if $1<p<\infty$, we prove that these sets are approximatively norm-compact. We introduce the class of uniformly approximable subsets of $L^p$, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the $p$-variation if $p\in [1,\infty )$ and in terms of covering numbers if $p=\infty$. 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 Hölder transformations.