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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hausdorff $ m$ regular and rectifiable sets in $ n$-space


Author: Pertti Mattila
Journal: Trans. Amer. Math. Soc. 205 (1975), 263-274
MSC: Primary 28A75
MathSciNet review: 0357741
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Abstract: The purpose of this paper is to prove the following theorem: If $ E$ is a subset of Euclidean $ n$-space and if the $ m$-dimensional Hausdorff density of $ E$ exists and equals one $ {H^m}$ almost everywhere in $ E$, then $ E$ is countably $ ({H^m},m)$ rectifiable. Here $ {H^m}$ is the $ m$-dimensional Hausdorff measure. The proof is a generalization of the proof given by J. M. Marstrand in the special case $ n = 3,m = 2$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0357741-4
PII: S 0002-9947(1975)0357741-4
Keywords: Hausdorff measure, density, regular set, rectifiable set
Article copyright: © Copyright 1975 American Mathematical Society