The integral formula for calculating the Hausdorff measure of some fractal sets (Q2744434)

Let \(F \subseteq [a, b]\) be a closed set in \({\mathbb R}\) and let \(\mu\) be a finite Borel measure carried by \(F\). For every \(x \in F\), define NEWLINE\[NEWLINE {\underline D}_{\mu}{\mathcal H}^s (x)= \lim_{\delta \to 0} \inf \biggl\{ {{|I|^s}\over {\mu(I)}} \biggr\},NEWLINE\]NEWLINE where the infimum is taken over all subintervals of \([a, b]\) containing \(x\) with \(|I|\leq \delta.\) In this paper under review, the authors prove that if \({\underline D}_{\mu}{\mathcal H}^s (x)< \infty\) for all \(x \in F\), then NEWLINE\[NEWLINE {\mathcal H}^s(F) = \int_F {\underline D}_{\mu}{\mathcal H}^s (x) d\mu (x), NEWLINE\]NEWLINE where \({\mathcal H}^s\) denotes the \(s\)-dimensional Hausdorff measure. As an example, the authors show that for a perturbed Cantor set \(F\) with \(\dim F=s\), \({\mathcal H}^s(F)= 1.\)