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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
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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.


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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

DOI: https://doi.org/10.1090/proc/14372
Keywords: Homogeneous forcing notion, idealized forcing, Hausdorff measure, $\sigma$-finite, Cicho\'n 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