%0 Journal Article
%T Visible and Invisible Cantor sets
%A Carlos Cabrelli
%A Udayan Darji
%A Ursula Molter
%J Mathematics
%D 2011
%I arXiv
%X 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.
%U http://arxiv.org/abs/1109.1174v1