Set-theoretical problems concerning Hausdorff measures

Abstract:

We show that the $\sigma$-ideal of Borel subsets of $\mathbb {R}^3$ of $\sigma$-finite 2-dimensional Hausdorff measure is not homogeneous. This partially answers a question of Zapletal.

We prove that each of the statements $\operatorname {cov}(\mathcal {N}) < \operatorname {cov}(\mathcal {N}^1_2)$, $\operatorname {cov}(\mathcal {N}^1_2) < \operatorname {non}(\mathcal {M})$, and $\operatorname {cov}(\mathcal {M}) < \operatorname {non}(\mathcal {N}^1_2)$ is consistent, where $\mathcal {N}^1_2$ is the $\sigma$-ideal of sets in the plane of 1-dimensional Hausdorff measure zero, and $\mathcal {N}$ and $\mathcal {M}$ are the usual null and meagre $\sigma$-ideals. This answers a question of Fremlin and settles the question of strictness of all the inequalities once we fit the cardinal invariants of $\mathcal {N}^1_2$ into the Cichoń diagram.

We prove that it is 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. This answers a question of Humke and Laczkovich.