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Weak*-convergence of Monge-Ampère measures. (English) Zbl 1106.32024
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 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 $$(dd^cg_j)^n\overset{\text{weak}^\ast}\longrightarrow(dd^cu)^n$$.

##### MSC:
 32U05 Plurisubharmonic functions and generalizations
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##### References:
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