Let (X,d,μ) be a space of homogeneous type. We study the relationship between two types of s-sets: relative to a distance and relative to a measure. We find a condition on a closed subset F of X under which F is an s-set relative to the measure μ if and only if F is an s-set relative to δ. Here δ denotes the quasi-distance defined by Macías and Segovia such that (X,δ,μ) is a normal space. In order to prove this result, we prove a covering type lemma and a type of Hausdorff measure based criterion for a given set to be an s-set relative to μ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-2-4,
author = {Marilina Carena and Marisa Toschi},
title = {On s-sets in spaces of homogeneous type},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {193-203},
zbl = {1315.28001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-2-4}
}
Marilina Carena; Marisa Toschi. On s-sets in spaces of homogeneous type. Colloquium Mathematicae, Tome 139 (2015) pp. 193-203. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-2-4/