Real Analysis Exchange

Hausdorff measures of different dimensions are isomorphic under the continuum hypothesis.

Márton Elekes

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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.

Article information

Real Anal. Exchange, Volume 30, Number 2 (2004), 605 - 616 .

First available in Project Euclid: 15 October 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A78: Hausdorff and packing measures
Secondary: 26A16: Lipschitz (Hölder) classes 26A45: Functions of bounded variation, generalizations 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]

Hausdorff measure isomorphism $CH$ Hausdorff dimension H\"older continuous bounded variation


Elekes, Márton. Hausdorff measures of different dimensions are isomorphic under the continuum hypothesis. Real Anal. Exchange 30 (2004), no. 2, 605 -- 616.

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