On the equality of Hausdorff measure and Hausdorff content

TL;DR

This paper investigates when Hausdorff measure equals Hausdorff content for self-similar sets, proving equality under certain conditions and exploring related measure properties and limitations.

Contribution

It establishes conditions for Hausdorff measure and content equality in self-similar sets and provides examples where this equality fails in more general fractal sets.

Findings

Equality holds for subsets of certain self-similar sets.

Hausdorff measure and content differ in non-self-similar fractals.

Results have implications for Ahlfors regularity.

Abstract

We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph-directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. For example, it fails in general for self-conformal sets, self-affine sets and Julia sets. We also give applications of our results concerning Ahlfors regularity. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and $\delta$-approximate packing pre-measure coincide for sufficiently small $\delta>0$.