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arXiv · 1705.00858

Exact Hausdorff measures of Cantor sets

Malin Palö Forsström
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Abstract

Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try to resolve this problem by studying an extension of the Hausdorff measures \( μ_h\) on \(\mathbb{R}\), allowing gauge functions to depend on the midpoint of the covering intervals instead of only on the diameter. As a main result, a theorem about the Hausdorff measure of any regular enough Cantor set, with respect to a chosen gauge function, is obtained.

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Malin Palö Forsström. 2017-05-02. Exact Hausdorff measures of Cantor sets. https://doi.org/10.14321/realanalexch.39.2.0367

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