Multifractal variation measures and multifractal density theorems.
For \(E\subseteq {\mathbf R}^d, q, t\in {\mathbf R} \) and \(\delta >0\), \(\mathcal H_\mu^{q,t}\) and \(\mathcal P_\mu^{q,t}\) denote, respectively, the multifractal Hausdorff measure and multifractal packing measure introduced by Olsen and Peyrière. In this paper the authors show that \(\mathcal H_\mu^{q,t}\) and \(\mathcal P_\mu^{q,t}\) can be expressed as Henstock-Thomson ``variation'' measures. That is, Thomson's fine variation (measure) defined by the variation function \(h(x,r) =1_E(x) \mu(B(x,r))^q (2r)^t \) for \(E\subseteq {\mathbf R}^d, q, t\in {\mathbf R}\) and a Borel probability measure \(\mu\) on \({\mathbf R}^d\) with \(\limsup_{r\to 0} \sup_x\frac{\mu(B(x,2r))}{\mu(B(x,r))}< \infty\) coincides with the multifractal Hausdorff measure \(\mathcal H_\mu^{q,t}\) and Thomson's full variation (measure) defined by the variation function \(h(x,r) =1_E(x) \mu(B(x,r))^q (2r)^t \) for \(E\subseteq {\mathbf R}^d, q, t\in {\mathbf R}\) and a Borel probability measure \(\mu\) on \({\mathbf R}^d\) coincides with the multifractal packing measure \(\mathcal P_\mu^{q,t}\), where \(1_E\) denotes the indicator function on \(E\) and \(B(x,r)\) denotes a ball in \({\mathbf R}^d\). As an application they prove a density theorem for the multifractal measures \(\mathcal H_\mu^{q,t}\) and \(\mathcal P_\mu^{q,t}\).
scientific article; zbMATH DE number 2084055