Subsets with finite measure of multifractal Hausdorff measures. (English) Zbl 0963.28004

Huang, Lihu; Yu, Jinghu

J. Math. Res. Expo. 20, No. 2, 166-170 (2000).

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

Keywords:

measure; Borel probability measure; multifractal Hausdorff measure; Federer condition; finite-measure subsets