Measurable Envelopes, Hausdorff Measures and Sierpi\'nski Sets
M\'arton Elekes

TL;DR
This paper explores the independence of the existence of measurable envelopes and Sierpiński sets with respect to Hausdorff measures in -dimensional space, revealing their independence from ZFC axioms.
Contribution
It demonstrates the independence of measurable envelopes and the consistency of Sierpi4ki sets being Hausdorff-measurable within set theory.
Findings
Measurable envelopes' existence is independent of ZFC.
Sierpi4ki sets can be consistent with Hausdorff measure measurability.
Results depend on set-theoretic assumptions beyond ZFC.
Abstract
We show that the existence of measurable envelopes of all subsets of with respect to the -dimensional Hausdorff measure is independent of . We also investigate the consistency of the existence of Sierpi\'nski sets measurable with respect to the -dimensional Hausdorff measure.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Topological and Geometric Data Analysis
