Measurable Envelopes, Hausdorff Measures and Sierpi\'nski Sets

We show that the existence of measurable envelopes of all subsets of ${\text{RR}}^n$ 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.