Visisble and Invisible Cantor Sets

CABRELLI, CARLOS A.; DARJI, UDAYAN; MOLTER, URSULA M.

Excursions in Harmonic Analysis

Birkhauser

Año: 2013; p. 11 - 21

In this article we study for which Cantor sets there exists a gaugefunction 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 µ for which the set has positive and finite µ-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 0 or non-σ-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.