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Dominated sets, microscopic sets and Hausdorff measures

Ondřej Zindulka, Piotr Nowakowski

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.

Dominated sets, microscopic sets and Hausdorff measures

Abstract

Let be a family of sequences of positive numbers that decrease to 0, let be a metric space and . is said to be -dominated if, for every , a countable cover of can be found such that for all . We examine the family of all -dominated sets, denoted by . In particular, we examine the connections between and families of sets with zero Hausdorff measure for some gauges.
Paper Structure (9 sections, 45 theorems, 65 equations)

This paper contains 9 sections, 45 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Dominated sets
  3. Hausdorff measures
  4. Dominated sets versus Hausdorff measure zero I
  5. Dominated sets versus Hausdorff measure zero II
  6. Additively complete sets
  7. Hausdorff dimension
  8. Appendix: Gauges
  9. Remarks and questions

Key Result

Theorem 2.4

Let $S\subseteq\mathbb{S}$.

Theorems & Definitions (87)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7: invariants2
  • Lemma 2.8
  • proof
  • ...and 77 more