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 R(2). We find that for 0 < s < t < 2, q < 1, then there exist a set E subset of R(2) and a probability measure mu on R(2) such that dim(mu,H)(,)(q) H(E) = s, dim(mu,P)(q) (E) = t and, in addition, H(mu)(q,s)vertical bar(E) similar to P(mu)(q,t)|(E).