Hausdorff 𝑚 regular and rectifiable sets in 𝑛-space

Mattila, Pertti

doi:10.1090/S0002-9947-1975-0357741-4

Hausdorff $m$ regular and rectifiable sets in $n$-space
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Trans. Amer. Math. Soc. 205 (1975), 263-274 Request permission

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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