On countable determination of the Kuratowski measure of noncompactness

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 ${\rm co}(B_0)$.