Convergence in capacity (Q2888776)

Let \(\mathcal E\) be the Cegrell class of plurisubharmonic functions (the largest class where the complex Monge-Ampère operator is well-defined and continuous under monotone limit transitions). It is shown that if a sequence of functions \(u_s\in\mathcal E\) converges to \(u\in\mathcal E\) in \(C_{n-1}\)-capacity and satisfies \(u_s\geq u_0\) for some \(u_0\in\mathcal E\), then \((dd^cu_s)^n\to (dd^cu)^n\). The example NEWLINE\[NEWLINEu_s=\max\Big\{{1\over s}\log|z|, s\log|w|\Big\}NEWLINE\]NEWLINE shows that the assumption on a common minorizer is essential.