On countable determination of the Kuratowski measure of noncompactness

TL;DR

This paper proves that for the Kuratowski measure of noncompactness in a Banach space, a bounded set's measure can be determined by a countable subset, answering a long-standing open question.

Contribution

It establishes that every bounded set in a Banach space has a countable subset with the same Kuratowski measure, using ultrapower techniques and strong finite representability.

Findings

Countable subsets can determine the Kuratowski measure of noncompactness.

Every bounded set is strongly finitely representable in a countable subset.

Ultrafilter methods relate the set to ultrapower constructions.

Abstract

A long-standing question in the theory of measures of noncompactness is that for the Kuratowski measure of noncompactness $\alpha$ defined on a metric space $M$, and for every bounded subset $B\subset M$, is there a countable subset $B_0\subset B$ such that $\alpha(B_0)=\alpha(B)$? In this paper, we give an affirmative answer to the question above. It is done by showing that for each nonempty set $B$ of a Banach space, there is a countable subset $B_0\subset B$ so that $B$ is strongly finitely representable in $B_0$, and that there is a free ultrafilter $\mathcal U$ so that $B$ is affinely isometric to a subset of the ultrapower $[{\rm co}(B_0)]_\mathcal U$ of ${\rm co}(B_0)$.