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2004-2005 Hausdorff measures of different dimensions are isomorphic under the continuum hypothesis.
Márton Elekes
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Real Anal. Exchange 30(2): 605-616 (2004-2005).


We show that the Continuum Hypothesis implies that for every $ 0 < d_1 \leq d_2 < n $ the measure spaces $(\mathbb{R}^n,\mathcal{M}_{\mathcal{H}^{d_1}},\mathcal {H}^{d_1})$ and $(\mathbb{R}^n,\mathcal{M}_{\mathcal{H}^{d_2}},\mathcal{H}^{d_2})$ are isomorphic, where $\mathcal{H}^d$ is $d$-dimensional Hausdorff measure and $\mathcal{M}_{d}$ is the $\sigma$-algebra of measurable sets with respect to $\mathcal{H}^d$. 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 Hölder continuous (or is of bounded variation) on a set of positive Hausdorff dimension.


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Márton Elekes. "Hausdorff measures of different dimensions are isomorphic under the continuum hypothesis.." Real Anal. Exchange 30 (2) 605 - 616, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 15 October 2005

zbMATH: 1106.28002
MathSciNet: MR2177422

Primary: 28A78
Secondary: 03E50 , 26A16 , 26A45 , 28A05

Keywords: $CH$ , Bounded variation , H\"older continuous , Hausdorff dimension , Hausdorff measure , isomorphism

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 2 • 2004-2005
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