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We show that the Continuum Hypothesis implies that for every $0<d_1\leqd_2<n$ the measure spaces $(\RR^n,\iM_{\iH^{d_1}},\iH^{d_1})$ and$(\RR^n,\iM_{\iH^{d_2}},\iH^{d_2})$ are isomorphic, where $\iH^d$ is$d$-dimensional Hausdorff measure and $\iM_{\Hd}$ is the $\sigma$-algebra ofmeasurable sets with respect to $\Hd$. This is motivated by the well-knownquestion (circulated by D. Preiss) whether such an isomorphism exists if wereplace measurable sets by Borel sets.  We also investigate the related question whether every continuous function(or the typical continuous function) is H\"older continuous (or is of boundedvariation) on a set of positive Hausdorff dimension.

Hausdorff measures of different dimensions are isomorphic under the Continuum Hypothesis