Measurable Envelopes, Hausdorff Measures and Sierpi\'nski Sets

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 Sierpi
4ki sets being Hausdorff-measurable within set theory.

Findings

Measurable envelopes' existence is independent of ZFC.

Sierpi
4ki 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 $\RR^n$ with respect to the $d$-dimensional Hausdorff measure $(0<d<n)$ is independent of $ZFC$. We also investigate the consistency of the existence of Sierpi\'nski sets measurable with respect to the $d$-dimensional Hausdorff measure.