The Hausdorff dimension and measure of some Cantor sets (Q1852335)

Let \(M\) be a compact interval on the real line and let \(E\) be a Borel subset of \(M\). For any Radon measure \(\mu\) on \(M\), define \[ \underline{D}_{\mu}{\mathcal H}^s(x) = \lim_{\delta \to 0} \inf {{|I|^s}\over {\mu(I)}}, \] where the infimum is taken over all closed intervals \(I\) containing \(x\) and \(0< |I|< \delta\). The author proves that if \(\underline{D}_{\mu}{\mathcal H}^s(x) < \infty\) for all \(x \in E\), then \[ {\mathcal H}^s(E) = \int_E \underline{D}_{\mu}{\mathcal H}^s(x) d\mu(x). \] Applications of this result to various Cantor-type sets are given.