On packing measures and a theorem of Besicovitch

TL;DR

This paper investigates the differences between Hausdorff and packing measures, demonstrating that a property holding for Hausdorff measures does not necessarily hold for packing measures, and explores related measure-theoretic questions.

Contribution

The paper proves that Besicovitch's null set property for Hausdorff measures does not extend to packing measures and examines related measure properties.

Findings

Null sets for Hausdorff measure are not necessarily null for packing measure.

The property does not hold for non-$\sigma$-finite packing measures.

Results extend to pre-packing measures.

Abstract

Besicovitch showed that if a set is null for the Hausdorff measure associated to a given dimension function, then it is still null for the Hausdorff measure corresponding to a smaller dimension function. We prove that this is not true for packing measures. Moreover, we consider the corresponding questions for sets of non-$\sigma$-finite packing measure, and for pre-packing measure instead of packing measure.