TITLE: Compactness properties of plurisubharmonic functions
QUESTION [7 upvotes]: I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.
Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $F$ be a family of plurisubharmonic functions on $\Omega.$  We may assume that these functions are continuous or smooth if we wish; also, I'm particularly interested in $n=2.$ 
Q1. In what convergence can we get a converging subsequence to a plurisubharmonic function? 
Q2. Are there further special conditions on the family $F$ that ensure compactness in stronger sense?

REPLY [3 votes]: Besides the above-mentioned results in various topologies, the following is quite useful:
Let $\Omega$ be an open subset of $\mathbb{C}^n$. If $\Omega$ is connected and $u_j, j \in \mathbb{N}$  is a decreasing sequence of PSH functions, then $u = \lim_{j \to \infty}u_j \in PSH(\Omega)$ or $u \equiv -\infty$. This is Theorem 2.9.14 (ii) in: M. Klimek: Pluripotential Theory, Clarendon Press 1991. It also appears as Proposition I.3 (iii) in the Appendix I by Russakovskii and Favorov to the Russian translation of P. Lelong, L. Gruman: Entire functions of several complex variables, Springer-Verlag 1986.
The result mentioned by Alex Eremenko can also be found in the main body of Lelong and Gruman's book, as  Theorem 1.27 (in a slightly more general form, for upper-semicontinuous regularization of the upper limit (in the sense of filters) of a locally bounded from above family of PSH functions).
The conspiracy is vast indeed, but it seems to be limited to  authors of texts in complex analysis.