Hausdorff measures of different dimensions are isomorphic under the Continuum Hypothesis

TL;DR

Under the Continuum Hypothesis, the paper proves that Hausdorff measure spaces of different dimensions are isomorphic, and explores the properties of continuous functions on sets of positive Hausdorff dimension.

Contribution

It demonstrates the isomorphism of Hausdorff measure spaces of different dimensions under CH and investigates the regularity of continuous functions on such sets.

Findings

Hausdorff measure spaces of different dimensions are isomorphic under CH

Question of isomorphism with Borel measurable sets remains open

Analysis of continuous functions on sets with positive Hausdorff dimension

Abstract

We show that the Continuum Hypothesis implies that for every $0<d_1\leq d_2<n$ 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 of measurable sets with respect to $\Hd$. This is motivated by the well-known question (circulated by D. Preiss) whether such an isomorphism exists if we replace 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 bounded variation) on a set of positive Hausdorff dimension.