On s-sets in spaces of homogeneous type

Abstract

Let $(X,d,\mu)$ be a space of homogeneous type. In this note 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 we have that $F$ is $s$-set relative to the measure $\mu$ if and only if $F$ is $s$-set relative to $\delta$. Here $\delta$ denotes the quasi-distance defined by Mac\'ias and Segovia such that $(X,\delta,\mu)$ is a normal space. In order to prove this result, we show a covering type lemma and a type of Hausdorff measure based criteria for the $s$-set condition relative to $\mu$ of a given set.