Measurable Envelopes, Hausdorff Measures and Sierpiński Sets

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 Sierpiński sets measurable with respect to the $d$-dimensional Hausdorff measure.

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