Visible and Invisible Cantor Sets

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

Excursions in Harmonic Analysis

Birkhäuser

Año: 2013; p. 11 - 23

In this chapter we study for which Cantor sets there exists a {em 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 {em 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 $0$ 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 $X$.