Weak*-convergence of Monge-Ampère measures (Q2509048)

For an open set \(\varOmega\subset\mathbb C^n\), let \(\mathcal{E}_0(\varOmega)\) be the family of all bounded functions \(\psi\in\mathcal{PSH}(\varOmega)\) such that \(\lim_{z\to\xi}\psi(z)=0\), \(\xi\in\partial\varOmega\), and \(\int_\varOmega(dd^c\psi)^n<+\infty\). Let, further, \(\mathcal{F}(\varOmega)\) denote the family of all bounded functions \(\varphi\in\mathcal{PSH}(\varOmega)\) such that there exists a sequence \((\varphi_j)_{j=1}^\infty\subset\mathcal{E}_0(\varOmega)\) with \(\varphi_j\searrow\varphi\) and \(\sup_j\int_\varOmega(dd^c\varphi_j)^n<+\infty\). Finally, let \(\mathcal{E}(\varOmega)\) be the family of all functions \(u\) such that for every relatively compact open set \(\omega\Subset\varOmega\) there exists a \(u_\omega\in\mathcal{F}(\varOmega)\) such that \(u=u_\omega\) in \(\omega\). The author shows that if \(\mathcal{F}(\varOmega)\ni u_j\overset{\text{weak}} \longrightarrow u\in\mathcal{F}(\varOmega)\) and \(\int_\varOmega v(dd^cu_j)^n\longrightarrow\int_\varOmega v(dd^cu)^n\) for a strictly plurisubharmonic function \(v\in\mathcal{E}_0(\varOmega)\), then \((dd^cu_j)^n\overset{\text{weak}^\ast}\longrightarrow(dd^cu)^n\). Using this result and a theorem by \textit{E. Poletsky} [Trans. Am. Math. Soc. 355, 1579--1591 (2003; Zbl 1023.32020)], the author proves the following fundamental theorem. Let \(\varOmega\subset\mathbb C^n\) be strongly hyperconvex, \(u\in\mathcal{PSH}^-(\varOmega)\cap L^1(\varOmega)\). Then there exists a sequence \((g_j)_{j=1}^\infty\) of multipole Green functions such that \(g_j\overset{L^1}\longrightarrow u\). Moreover, if \(u\in\mathcal{E}\), then the sequence \((g_j)_{j=1}^\infty\) may be chosen so that \((dd^cg_j)^n\overset{\text{weak}^\ast}\longrightarrow(dd^cu)^n\).