  We present upper estimates for the number of negative eigenvalues of
two-dimensional Schroedinger operators with potentials generated by Ahlfors
regular measures of arbitrary dimension $\alpha\in (0, 2]$.The estimates are
given in terms of the integrals of the potential with a logarithmic weight and
of its L $\log$ L type Orlicz norms. In the case $\alpha = 1$, our estimates
are stronger than the known ones about Schroedinger operators with potentials
supported by Lipschitz curves.
