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The Hausdorff measure due to Davies and Rogers and its cardinal invariants

Tatsuya Goto

TL;DR

The paper analyzes the cardinal invariants of Davies and Rogers’ $f$-Hausdorff measure, which assigns either $0$ or $\infty$ to every Borel set. It establishes a general framework linking $f$-Hausdorff null sets to localization/anti-localization cardinals and derives ZFC inequalities tying these invariants to the classical ideals $\mathcal{N}$, $\mathcal{E}$, and $\mathcal{M}$. It then develops forcing constructions (including Zapletal-style iterations and Klausner–Mejía slalom forcing) to separate key invariants, proving consistency of inequalities such as $\operatorname{cov}(\mathcal{E}) < \operatorname{non}(\mathcal{N}^h_\Omega)$ and $\operatorname{cov}(\mathcal{N}^h_\Omega) < \operatorname{non}(\mathcal{E})$ in various models. The results illuminate how the Davies–Rogers measure interacts with localization cardinals and standard set-theoretic invariants, and they raise natural open questions about further separations and inequalities. Overall, the work advances understanding of the descriptive-set-theoretic landscape surrounding measures with extreme additivity properties and their cardinal characteristics.

Abstract

Davies and Rogers constructed a Hausdorff measure satisfying the following property: every Borel subset of the space has measure either $\infty$ or $0$. In this paper, we examine cardinal invariants of their measure.

The Hausdorff measure due to Davies and Rogers and its cardinal invariants

TL;DR

The paper analyzes the cardinal invariants of Davies and Rogers’ -Hausdorff measure, which assigns either or to every Borel set. It establishes a general framework linking -Hausdorff null sets to localization/anti-localization cardinals and derives ZFC inequalities tying these invariants to the classical ideals , , and . It then develops forcing constructions (including Zapletal-style iterations and Klausner–Mejía slalom forcing) to separate key invariants, proving consistency of inequalities such as and in various models. The results illuminate how the Davies–Rogers measure interacts with localization cardinals and standard set-theoretic invariants, and they raise natural open questions about further separations and inequalities. Overall, the work advances understanding of the descriptive-set-theoretic landscape surrounding measures with extreme additivity properties and their cardinal characteristics.

Abstract

Davies and Rogers constructed a Hausdorff measure satisfying the following property: every Borel subset of the space has measure either or . In this paper, we examine cardinal invariants of their measure.
Paper Structure (8 sections, 17 theorems, 55 equations)

This paper contains 8 sections, 17 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Review of Hausdorff measures
  3. General results on additivity and cofinality
  4. Review of the construction due to Davies and Rogers
  5. $\mathsf{ZFC}$ results
  6. Concictency Results
  7. Consistency of $\operatorname{cov}(\mathcal{E}) < \operatorname{non}(\mathcal{N}^h_\Omega)$
  8. Questions

Key Result

Lemma 7

Let $(X,d)$ be a separable metric space. Then there exists a countable family $\mathcal{C}$ of subsets of $X$ such that for every $A\subseteq X$ of finite diameter and every $\varepsilon>0$, there is some $C\in\mathcal{C}$ with Moreover, if $X$ is also an ultrametric space, then we can strengthen this to

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Remark 6
  • Lemma 7
  • proof
  • Corollary 8
  • proof
  • ...and 25 more