On packing measures and a theorem of Besicovitch

Garcia, Ignacio; Shmerkin, Pablo

doi:10.1090/S0002-9939-2014-11962-9

On packing measures and a theorem of Besicovitch
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by Ignacio Garcia and Pablo Shmerkin

Proc. Amer. Math. Soc. 142 (2014), 2661-2669

DOI: https://doi.org/10.1090/S0002-9939-2014-11962-9

Published electronically: April 24, 2014

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Abstract:

Let $\mathcal {H}^h$ be the $h$-dimensional Hausdorff measure on $\mathbb {R}^d$. Besicovitch showed that if a set $E$ is null for $\mathcal {H}^h$, then it is null for $\mathcal {H}^g$, for some dimension $g$ smaller than $h$. 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.

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