  Given an elliptic operator~$L$ on a bounded domain~$\Omega \subseteq {\bf
R}^n$, and a positive Radon measure~$\mu$ on~$\Omega$, not charging polar sets,
we discuss an explicit approximation procedure which leads to a sequence of
domains~$\Omega_h \subseteq \Omega$ with the following property: for
every~$f\in H^{-1}(\Omega)$ the sequence~$u_h$ of the solutions of the
Dirichlet problems~$L\, u_h=f$ in~$\Omega_h$, $u_h=0$ on~$\partial \Omega_h$,
extended to 0 in~$\Omega \setminus \Omega_h$, converges to the solution of the
\lq\lq relaxed Dirichlet problem\rq\rq\ $L\,u+\mu u=f$ in~$\Omega$, $u=0$
on~$\partial \Omega$.
