On packing measures and a theorem of Besicovitch

Abstract

Let H h \mathcal {H}^h be the h h -dimensional Hausdorff measure on R d \mathbb {R}^d . Besicovitch showed that if a set E E is null for H h \mathcal {H}^h , then it is null for H g \mathcal {H}^g , for some dimension g g smaller than h h . We prove that this is not true for packing measures. Moreover, we consider the corresponding questions for sets of non- σ \sigma -finite packing measure and for pre-packing measure instead of packing measure.