Visible and Invisible Cantor sets

TL;DR

This paper investigates the measure-theoretic properties of Cantor sets, showing that certain measure conditions are dense among all compact subsets of Polish spaces, and explores examples of dimensionless Cantor sets with unique measure characteristics.

Contribution

It demonstrates the density of Cantor sets with positive finite Hausdorff measure and introduces the concept of dimensionless Cantor sets with special measure properties.

Findings

The collection of Cantor sets with positive finite Hausdorff measure is dense.

Generic Cantor sets admit translation-invariant measures with positive finite measure.

Dimensionless Cantor sets form a dense subset among compact sets in Polish spaces.

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.