On countable determination of the Kuratowski measure of noncompactness

[摘要] 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 of M, is there a countable subset B-0 subset of 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 of B so that B is strongly finitely representable in B-0, and that there is a free ultrafilter U so that B is affinely isometric to a subset of the ultrapower [co(B-0)](u) of co(B-0). (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Kuratowski's measure of noncompactness;Countable determination;Strongly finite representability;Banach space;Metric space