Upper envelope of plurisubharmonic functions Suppose that $\\{u_\alpha\\}_{\alpha \in A}$ is a family of plurisubharmonic functions (psh function) on $\Omega \subset\subset \mathbb{C}^n$. Then, let $u(z) = \sup_{\alpha \in A} u_{\alpha}(z)$ be the upper envelope. I found in lots of literature that $$u^\ast(z) := \lim_{\epsilon \rightarrow 0} \sup_{B_{\epsilon}(z)} u(z)$$ (usc regularization) is still a psh fucntion. Is there any counterexample that the upper envelope $u$ for psh functions $u_\alpha$ before doing usc regulatization is not psh, I wonder?

Here is an example that works already for $n=1$. Let $\Omega$ be the unit disc and let $$ u_n(z) = \frac1n \log|z|. $$ Then $$ u(z) = \sup_n u_n(z) = \begin{cases} 0, & z \
eq 0 \\\ -\infty, & z = 0, \end{cases} $$ which is not upper semicontinuous, and therefore not (pluri-)subharmonic.

It may also be helpful to know that $u$ is "almost" psh already before the usc regularization; assuming that the family $\\{ u_\alpha \\}$ is locally upper bounded, then $u = u^*$ outside a pluripolar set, so in particular outside a set of Lebesgue measure $0$.