A note on approximation of plurisubharmonic functions

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\mathrm{\Omega }$ and continuous on $\bar{\mathrm{\Omega }}$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\bar{\mathrm{\Omega }}$. We show that such approximation is possible if the boundary of $\mathrm{\Omega }$ is $C^0$ outside a countable exceptional set $E\subset \mathrm{\partial }\mathrm{\Omega }$. In particular, approximation is possible on the Hartogs triangle. For H"older continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximation is not possible, even when approximation in the H"older continuous case is possible.