Measurability of multifractal measure functions and multifractal dimension functions
type: Item
instance of: scholarly article
title: Measurability of multifractal measure functions and multifractal dimension functions
zbMATH Open document ID: 0982.28003
author: L. Olsen
published in: Hiroshima Mathematical Journal
publication date: 2002-03-29T00:00:00Z
review text: 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.
reviewed by: Y. Xiao
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1423813
zbMATH Keywords: multifractals
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: packing measure
zbMATH Keywords: packing dimension
zbMATH Keywords: Baire hierarchy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206124851
OpenAlex ID: W1547356560
Wikidata QID: Q128870139
DOI: 10.32917/HMJ/1206124851
description: scientific article; zbMATH DE number 1423813