Dominated sets, microscopic sets and Hausdorff measures

Abstract

Let $S$ be a family of sequences of positive numbers that decrease to 0, let $X$ be a metric space and $A\subset X$. $A$ is said to be $S$-dominated if, for every $s\in S$, a countable cover $\{E_n\}$ of $E$ can be found such that $diam{E}_n<s_n$ for all $n$. We examine the family of all $S$-dominated sets, denoted by $\mathcal{D}(S)$. In particular, we examine the connections between $\mathcal{D}(S)$ and families of sets with zero Hausdorff measure for some gauges.