On the multifractal analysis of measures in a probability space

Abstract

In this paper, we calculate the relative multifractal Hausdorff and packing dimensions of measures in a probability space. Also, we obtain the analogue of Frostman’s lemma in a probability space for a relative multifractal Hausdorff measure. In the same way, there is a valid result for the relative multifractal packing pre-measure. Furthermore, we obtain the representations of the functions b and B by means of the analogue of Frostman’s lemma, and we provide a technique for showing that E is a $(q,\mathrm{\mu })$-fractal with respect to ν. In addition, we suggest new proofs of theorems on the relative multifractal formalism in a probability space. They yield results even at a point q for which the multifractal functions $b(q)$ and $B(q)$ differ.