Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures

Abstract

Motivated by Bedford's question, we study the possible cluster sets, in the vague topology, of Monge--Amp\`ere measures associated with bounded plurisubharmonic sequences $u_j\to\varphi$ locally in $L^1$, without assuming a common bound. On a bounded domain $\Omega\subset\mathbb C^n$ ($n\geq2$), and for a bounded maximal plurisubharmonic $\varphi$, the possible sets of all vague subsequential measure limits are precisely the vaguely closed sets of nonnegative Radon measures, including the empty set. This conclusion holds for every bounded plurisubharmonic $\varphi$ when $\Omega$ is hyperconvex, pseudoconvex and Runge, or a domain obtained by removing a relatively closed pluripolar set from a bounded hyperconvex domain. Under these hypotheses, every nonnegative Radon measure $\nu$ can also be realized by such a sequence with $(dd^cu_j)^n\to\nu$ vaguely. The main analytic result holds on every bounded domain: for any bounded plurisubharmonic $\varphi$ and nonnegative Radon measure $\mu$, we construct $u_j\leq\varphi$ converging to $\varphi$ in $L^p(\Omega)$ for every $1\leq p<\infty$, with $(dd^cu_j)^n\to(dd^c\varphi)^n+\mu$ vaguely. The approximants can be chosen smooth when $\varphi$ is continuous. On arbitrary pseudoconvex domains, prescribed measure limits can also be realized using locally bounded approximants. Under an additional localization condition, the approximants can be chosen globally bounded when the limit function is bounded.