Theorem 5.27.36.CCB8860FB3B5 Let $\Omega \subset \mathbb C^ n$, $n\geq 2$, be a bounded domain, let $b\in \mathrm{PSH}(\Omega )\cap L^\infty (\Omega )$, and let $\nu $ be a positive Radon measure on $\Omega $. There are bounded plurisubharmonic functions $u_ j\leq b$ on $\Omega $ such that
\[ \begin{gathered} \| u_ j-b\| _{L^ p(\Omega )}\longrightarrow 0\quad (1\leq p<\infty ),\\ (\mathrm{dd}^{\mathrm{c}}u_ j)^ n\longrightarrow (\mathrm{dd}^{\mathrm{c}}b)^ n+\nu \quad \text{vaguely}. \end{gathered} \]
The same sequence converges to $b$ in local capacity. If $b$ is smooth, every $u_ j$ can be chosen smooth. The measure $\nu $ may have infinite total mass.
Proof.
Choose relatively compact open sets exhausting $\Omega $, each with closure contained in the next one. On each of these sets, take a countable family of smooth compactly supported functions that is uniformly dense in the space of continuous functions vanishing at its boundary, and add a nonnegative smooth compactly supported cutoff function equal to one on its closure. Enumerate the union of these families as $(f_ i)$. If positive measures converge against every $f_ i$ to a fixed positive Radon measure, they converge vaguely: the cutoff functions give common local mass bounds, and uniform approximation then handles every continuous compactly supported test function.
For each $j$, approximate $\nu $ against $f_1,\ldots ,f_ j$ by a finite positive atomic measure
\[ \nu _ j=\sum _{s=1}^{N_ j}c_{j,s}\delta _{a_{j,s}},\qquad \left|\int _\Omega f_ i(\nu _ j-\nu )\right|<j^{-1}\quad (i\leq j). \]
Such an approximation uses only the finite mass of $\nu $ on a compact neighborhood of the supports of these tests: a finite partition on which all the tests have small oscillation provides the atoms and their positive weights. Atoms of weight zero are omitted. If $N_ j=0$, set $u_ j=b$. Otherwise start with $b_{j,0}=b$ and apply proposition 5.27.35 successively to $b_{j,s-1}$, $a_{j,s}$ and $c_{j,s}$, with the tests $f_1,\ldots ,f_ j$ and $\tau =t=(jN_ j)^{-1}$. This gives perturbations $h_{j,s}$, and we define
\[ b_{j,s}=b_{j,s-1}+h_{j,s},\qquad u_ j=b_{j,N_ j}. \]
Every intermediate potential is bounded and plurisubharmonic, so the proposition applies at each step; the choice at the next step may depend on the bound of the current potential, and each perturbation serves all finite $p$ at once. Summing the norm estimates yields $\| u_ j-b\| _{L^ p(\Omega )}\leq C_{\Omega ,p}/j$, and since every $h_{j,s}\leq 0$, we have $u_ j\leq b$. Summing the measure errors gives
\[ \left|\int _\Omega f_ i\bigl ((\mathrm{dd}^{\mathrm{c}}u_ j)^ n-(\mathrm{dd}^{\mathrm{c}}b)^ n-\nu \bigr )\right| <2/j\quad (i\leq j). \]
By the choice of the family $(f_ i)$, this proves the asserted vague limit, also when $\nu (\Omega )=\infty $.
For fixed concentric balls $B'\Subset B\Subset \Omega $, the perturbations are nonpositive, so $|u_ j-b|=\sum _ s|h_{j,s}|$. The last estimate of proposition 5.27.35 bounds the integral of this function against $(\mathrm{dd}^{\mathrm{c}}w)^ n$ over $B'$ by $C_{B',B,\Omega }/j$, for every $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Restricting to the set where $|u_ j-b|$ exceeds a fixed threshold, dividing by the threshold and taking the supremum over $w$ proves convergence in local capacity. If $b$ is smooth, the finite sums defining $u_ j$ are smooth as well.
$\square$
Pome
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