Set-theoretical problems concerning Hausdorff measures

TL;DR

This paper investigates the homogeneity of certain Hausdorff measure-based forcing notions and explores the relationships between cardinal invariants of measure-zero sets, providing new consistency results and answering open questions.

Contribution

It shows that the $\sigma$-ideal of Borel sets with $\sigma$-finite 2D Hausdorff measure is not homogeneous and establishes new inequalities among cardinal invariants, answering open problems.

Findings

The $\sigma$-ideal of Borel sets with $\sigma$-finite 2D Hausdorff measure is not homogeneous.

Consistently, $ ext{cov}( ext{Hausdorff measure null sets}) > ext{cov}( ext{Lebesgue null sets})$.

The paper confirms the possibility of ordering reals with all proper initial segments Lebesgue null but not Hausdorff measure null.

Abstract

J. Zapletal asked if all the forcing notions considered in his monograph are homogeneous. Specifically, he asked if the forcing consisting of Borel sets of $\sigma$-finite 2-dimensional Hausdorff measure in $\mathbb{R}^3$ (ordered under inclusion) is homogeneous. We give a partial negative answer to both questions by showing that this $\sigma$-ideal is not homogeneous. Let $\mathcal{N}^1_2$ be the $\sigma$-ideal of sets in the plane of 1-dimensional Hausdorff measure zero. D. H. Fremlin determined the position of the cardinal invariants of this $\sigma$-ideal in the Cicho\'n Diagram. This required proving numerous inequalities, and in all but three cases it was known that the inequalities can be strict in certain models. For one of the remaining ones Fremlin posed this as an open question in his monograph. We answer this by showing that consistently $\mathrm{cov}(\mathcal{N}^1_2) > \mathrm{cov}(\mathcal{N})$, where $\mathcal{N}$ is the usual Lebesgue null ideal. We also prove that the remaining two inequalities can be strict. Moreover, we fit the cardinal invariants of the $\sigma$-ideal of sets of $\sigma$-finite Hausdorff measure into the diagram. P. Humke and M. Laczkovich raised the following question. Is it consistent that there is an ordering of the reals in which all proper initial segments are Lebesgue null but for every ordering of the reals there is a proper initial segment that is not null with respect to the $1/2$-dimensional Hausdorff measure? We determine the values of the cardinal invariants of the Cicho\'n Diagram as well as the invariants of the nullsets of Hausdorff measures in the first model mentioned in the previous paragraph, and as an application we answer this question of Humke and Laczkovich affirmatively.