Abstract
We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H s) and (ℝ, B, H t) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss.
To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d.
We also prove this statement in a more general form: If A ⊂ ℝn is Borel and ƒ: A → ℝm is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A).
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References
M. Elekes, Hausdorff measures of different dimensions are isomorphic under the Continuum Hypothesis, Real Analysis Exchange 30 (2004/05), 605–616.
A. Máthé, Measurable functions are of bounded variation on a set of dimension 1/2, in preparation.
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Selected Problems on M. Csornyei’s homepage see site, http://www.homepages.ucl.ac.uk/~ucahmcs/probl.ps
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Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.
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Máthé, A. Hausdorff measures of different dimensions are not Borel isomorphic. Isr. J. Math. 164, 285–302 (2008). https://doi.org/10.1007/s11856-008-0030-5
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DOI: https://doi.org/10.1007/s11856-008-0030-5