Best approximation properties in spaces of measurable functions

TL;DR

This paper investigates approximation properties in measurable function spaces, focusing on proximinality, the Kadec-Klee property, and metric projection continuity, with new criteria and conditions established for various function spaces.

Contribution

It introduces new criteria for proximinality, explores the Kadec-Klee property in different spaces, and provides conditions for metric projection continuity and metric selection in Lorentz spaces.

Findings

Proximinality of certain sets in measurable function spaces characterized.

Equivalence between Kadec-Klee property and $mbda$-compactness established.

Necessary and sufficient conditions for metric projection continuity and selection derived.

Abstract

We research proximinality of $\mu$-sequentially compact sets and $\mu$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $\mu$-compactness of the sets in Banach function spaces. Also the property $S$ is investigated in Fr\'echet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fr\'echet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces $d(w,1)$.