Dominated sets, microscopic sets and Hausdorff measures

TL;DR

This paper explores the properties of $S$-dominated sets in metric spaces, analyzing their relationship with sets of zero Hausdorff measure and introducing new insights into geometric measure theory.

Contribution

It introduces the concept of $S$-dominated sets and investigates their connections with Hausdorff measures, expanding understanding of measure-theoretic properties of metric spaces.

Findings

Characterization of $S$-dominated sets in relation to Hausdorff measures

Identification of conditions under which $S$-dominated sets have zero Hausdorff measure

Establishment of links between dominated sets and measure-theoretic properties in metric spaces

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.