  We study the problem of approximating plurisubharmonic functions on a bounded
domain $\Omega$ by continuous plurisubharmonic functions defined on
neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to
the problem of solving a Dirichlet type problem for functions plurisubharmonic
on the compact set $\bar\Omega$ in the sense of Poletsky. A stronger notion of
hyperconvexity is introduced to fully utilize this connection, and we show that
for this class of domains the duality between the two problems is perfect. In
this setting, we give a characterization of plurisubharmonic boundary values,
and prove some theorems regarding the approximation of plurisubharmonic
functions.
