Set-theoretical problems concerning Hausdorff measures

Authors:
Márton Elekes and Juris Steprāns

Journal:
Proc. Amer. Math. Soc. **147** (2019), 1709-1717

MSC (2010):
Primary 03E35, 28A78, 03E17; Secondary 03E40, 03E75.

DOI:
https://doi.org/10.1090/proc/14372

Published electronically:
December 12, 2018

MathSciNet review:
3910435

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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**Márton Elekes**

Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary – and – Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117 Budapest, Hungary

Email:
elekes.marton@renyi.mta.hu

**Juris Steprāns**

Affiliation:
Department of Mathematics, York University, Toronto, Ontario M3J 1P3, Canada

Email:
steprans@mathstat.yorku.ca

Keywords:
Homogeneous forcing notion,
idealized forcing,
Hausdorff measure,
$\sigma$-finite,
Cichoń diagram,
linear ordering

Received by editor(s):
August 12, 2015

Received by editor(s) in revised form:
April 27, 2018

Published electronically:
December 12, 2018

Additional Notes:
The first author was partially supported by the Hungarian Scientific Foundation grants no. 83726, 104178, and 113047.

The second author was partially supported by a Discovery Grant from NSERC

This research was partially done whilst the authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences in the programme ‘Mathematical, Foundational and Computational Aspects of the Higher Infinite’ (HIF)

Communicated by:
Mirna Džamonja

Article copyright:
© Copyright 2018
American Mathematical Society