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Hausdorff measures of different dimensions are not Borel isomorphic
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Máthé, András (2008) Hausdorff measures of different dimensions are not Borel isomorphic. Israel Journal of Mathematics, Vol.164 (No.1). pp. 285-302. doi:10.1007/s11856-008-0030-5 ISSN 0021-2172.
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Official URL: http://dx.doi.org/10.1007/s11856-008-0030-5
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).
Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Israel Journal of Mathematics | ||||
Publisher: | Magnes Press | ||||
ISSN: | 0021-2172 | ||||
Official Date: | 2008 | ||||
Dates: |
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Volume: | Vol.164 | ||||
Number: | No.1 | ||||
Page Range: | pp. 285-302 | ||||
DOI: | 10.1007/s11856-008-0030-5 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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