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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
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