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Subsets with finite measure of multifractal Hausdorff measures. (English) Zbl 0963.28004
Summary: Let $$\mu$$ be a Borel probability measure on $$\mathbb{R}^d$$, $$q,t\in\mathbb{R}$$. Let $${\mathcal H}^{q,t}_\mu$$ denote the multifractal Hausdorff measure. We prove that, when $$\mu$$ satisfies the so-called Federer condition for a closed subset $$E\in\mathbb{R}^n$$ such that $${\mathcal H}^{q,t}_\mu(E)> 0$$, there exists a compact subset $$F$$ of $$E$$ with $$0<{\mathcal H}^{q,t}_\mu(F)< \infty$$, i.e., finite-measure subsets of multifractal Hausdorff measure exist.

##### MSC:
 28A78 Hausdorff and packing measures