Carleson measures and uniformly perfect sets

We show that the description of Carleson measures on the Bergman spase of analytic functions on a finitely connected domain $G$ with the power weight is the same one as in the unit disk iff the complement $\overline{\mathbb C}\setminus G$ be an unbounded set without isolated points. In general case the complement of such domain $G$ have to be a uniformly perfect set.