Measurability of multifractal measure functions and multifractal dimension functions
For a Radon measure \(\mu\) and \(q, t \in {\mathbf R}\), let \({\mathcal H}_{\mu}^{q, t}\) and \({\mathcal P}_{\mu}^{q, t}\) be, respectively, the multifractal Hausdorff measure and the multifractal packing measure introduced by the author [Adv. Math. 116, No. 1, 85-196 (1995; Zbl 0841.28012)]. In this paper, the author studies the descriptive set theoretic complexity of the maps \[ {\mathcal K}({\mathbf R}^d) \times {\mathcal M}({\mathbf R}^d) \times {\mathbf R} \to \overline {\mathbf R}: (K, \mu, q) \mapsto {\mathcal H}_{\mu}^{q, t}, \] \[ {\mathcal K}({\mathbf R}^d) \times {\mathcal M}({\mathbf R}^d) \times {\mathbf R} \to \overline {\mathbf R}: (K, \mu, q) \mapsto {\mathcal P}_{\mu}^{q, t}, \] and the related multifractal dimension maps; where \({\mathcal K}({\mathbf R}^d)\) denotes the family of non-empty compact sets of \({\mathbf R}^d\) equipped with the Hausdorff metric, and \({\mathcal M}({\mathbf R}^d) \) denotes the family of Radon measures on \({\mathbf R}^d\) equipped with the weak topology. The author proves that both Hausdorff measure and Hausdorff dimension functions are \(\sigma({\mathcal A})\)-measurable, where \(\sigma({\mathcal A})\) is the \(\sigma\)-algebra generated by the family \(\mathcal A\) of analytic sets of \({\mathcal K}({\mathbf R}^d) \times {\mathcal M}({\mathbf R}^d) \times {\mathbf R}\). These results are applied to prove measurability of multifractal slices. The results obtained for the multifractal packing measures are weaker. In particular, the problem whether the multifractal packing measure function is \(\sigma({\mathcal A})\)-measurable remains open.
scientific article; zbMATH DE number 1423813