Hausdorff measures, dimensions and mutual singularity

Abstract:

Let $(X,d)$ be a metric space. For a probability measure $\mu$ on a subset $E$ of $X$ and a Vitali cover $V$ of $E$, we introduce the notion of a $b_{\mu }$-Vitali subcover $V_{\mu }$, and compare the Hausdorff measures of $E$ with respect to these two collections. As an application, we consider graph directed self-similar measures $\mu$ and $\nu$ in $\mathbb {R}^{d}$ satisfying the open set condition. Using the notion of pointwise local dimension of $\mu$ with respect to $\nu$, we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen’s multifractal Hausdorff measures are mutually singular.