  In this paper, we investigate what are Carleson measures on open subsets in
the complex plane. A circular domain is a connected open subset whose boundary
consists of finitely many disjoint circles. We call a domain $G$ multi-nicely
connected if there exists a circular domain $W$ and a conformal map $\psi$ from
$W$ onto $G$ such that $\psi$ is almost univalent with respect the arclength on
$\partial W$. We characterize all Carleson measures for those open subsets so
that each of their components is multi-nicely connected and harmonic measures
of the components are mutually singular. Our results suggest the extend of
Carleson measures probably is up to this class of open subsets.
