On packing measures and a theorem of Besicovitch

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-$σ$-finite packing measure, and for pre-packing measure instead of packing measure.