Convergence in capacity
The author studies how the convergence in capacity (for Bedford-Taylor relative capacity \(C_n\) and the capacity \(C_{n-1}\) defined by \[ C_{n-1} (E)=\sup \biggl\{ \int _E (dd^c u)^{n-1} \wedge (dd^c | z| ^2 ); \;u\in \text{PSH}(\Omega ),\;0<u<1 \biggr\} \] for compact \(E\subset \Omega\subset \mathbb C ^n \)) is related to the convergence of Monge-Ampère measures \((dd^c u_j )^n \to (dd^c u )^n\) associated to plurisubharmonic functions \(u_j\). He proves that if for \(u_j \in \text{PSH}(\Omega )\) \(u_j \to u\) and \((dd^c u_j )^n \to (dd^c u )^n\) weakly, and (essentially) \(u_j \geq u\) on \(\partial \Omega\) then \(u_j \to u\) in \(C_{n-1}\) capacity. He also obtains some generalizations of a stability theorem from \textit{U. Cegrell} and \textit{S. Kołodziej} [Math. Ann. 334, No.~4, 713--729 (2006; Zbl 1103.32019)] and a convergence theorem of \textit{U. Cegrell} [I. Newton Inst. for Math. Sci., preprint series NI01046-NPD].
scientific article; zbMATH DE number 5315511