On the equivalence of the multifractal centred Hausdorff measure and the multifractal packing measure

In this work, we focus on the multifractal centred Hausdorff measure and the multifractal packing measure in R2. We find that for 0 < s < t < 2, q < 1, then there exist a set E ⊆ R2 and a probability measure μ on R2 such that dimμ,Hq(E) =s, dimμ,Pq(E) = t and, in addition, Hq,sμ|E∼Pq,tμ|E