On s-sets in spaces of homogeneous type

Let $(X,d,μ)$ 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 $μ$ if and only if $F$ is $s$-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 show a covering type lemma and a type of Hausdorff measure based criteria for the $s$-set condition relative to $μ$ of a given set.

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