Visible and Invisible Cantor sets

Carlos Cabrelli; Udayan Darji; Ursula Molter

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arXiv · 1109.1174

Visible and Invisible Cantor sets

Carlos Cabrelli·Udayan Darji·Ursula Molter

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Abstract

In this article we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff-measure is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set satisfies that there exists a translation-invariant measure mu for which the set has positive and finite mu-measure. In contrast, we generalize an example of Davies of dimensionless Cantor sets (i.e. a Cantor set for which any translation invariant measure is either zero or non-sigma-finite, that enables us to show that the collection of these sets is also dense in the set of all compact subsets of a Polish space X.

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BibTeX RIS

Carlos Cabrelli, Udayan Darji, Ursula Molter. 2011-09-06. Visible and Invisible Cantor sets. https://arxiv.org/abs/1109.1174

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