  We consider the continuous model of log-infinitely divisible multifractal
random measures (MRM) introduced in \cite{bacry} . If M is a non degenerate
multifractal measure with associated metric $\rho(x,y)=M([x,y])$ and structure
function $\zet a$, we show that we have the following relation between the
(Euclidian) Hausdorff dimension ${\rm dim}_H$ of a measurable set K and the
Hausdorff dimension ${\rm dim}_H^{\rho}$ with respect to \rho of the same set:
$\zeta({\rm dim}_H^{\rho}(K))={\r m dim}_H(K)$. Our results can be extended to
higher dimensions in the log normal case: inspired by quantum gravity in dime
nsion 2, we consider the 2 dimensional case.
