Theorem 5.27.39.DCCBB28F9C78 Let $\Omega \subset \mathbb C^ n$, $n\geq 2$, be a bounded domain. A set $F\subset \mathcal M_+(\Omega )$ is vaguely closed if and only if there are smooth bounded plurisubharmonic functions $u_ j\leq 0$ on $\Omega $, tending to zero in $L^ p(\Omega )$ for every $1\leq p<\infty $ and in local capacity, whose full vague cluster set is $F$:
\[ F=\left\{ \mu \in \mathcal M_+(\Omega ):\ (\mathrm{dd}^{\mathrm{c}}u_{j_ k})^ n\longrightarrow \mu \text{ vaguely for some }j_1<j_2<\cdots \right\} . \]
The empty set is allowed.
Proof.
In a metric space, the full cluster set of a sequence is the intersection of the closures of its tails: a point of this intersection can be approximated within $1/k$ at an index larger than the preceding one, successively for $k=1,2,\ldots $. The full cluster set is therefore closed, and lemma 5.27.38 proves necessity.
For sufficiency, suppose first that $F$ is nonempty and closed. A subset of a separable metric space is separable, so we may choose a dense sequence $\mu _1,\mu _2,\ldots $ in $F$, repetitions allowed. Concatenating the finite lists $(\mu _1)$, $(\mu _1,\mu _2)$, $(\mu _1,\mu _2,\mu _3)$, and so on, we obtain a sequence $(\nu _ j)$ in which every $\mu _ k$ appears arbitrarily late. Hence every point of $F$ is the limit of a subsequence, and since $F$ is closed, no other subsequential limit occurs. If $F$ is empty, take $\nu _ j=j\delta _ a$ for a fixed $a\in \Omega $. A nonnegative compactly supported test function equal to one at $a$ has integral $j$ against $\nu _ j$, so this sequence has no vaguely convergent subsequence.
In either case, let $(f_ i)$ and $d_ v$ be as in lemma 5.27.38, and perform the finite atomic construction from the proof of theorem 5.27.36 separately for each $j$, with background zero, target $\nu _ j$ and tests $f_1,\ldots ,f_ j$. Thus we approximate $\nu _ j$ in these coordinates by a finite atomic measure with error less than $1/j$. If this measure has $N_ j>0$ atoms, apply proposition 5.27.35 successively with $t=\tau =(jN_ j)^{-1}$; if it has none, take $u_ j=0$. The resulting smooth bounded plurisubharmonic functions $u_ j\leq 0$ satisfy
\[ \| u_ j\| _{L^ p(\Omega )}\leq C_{\Omega ,p}/j, \qquad \left|\int _\Omega f_ i\bigl ((\mathrm{dd}^{\mathrm{c}}u_ j)^ n-\nu _ j\bigr )\right| <2/j\quad (i\leq j). \]
On every fixed pair of concentric balls $B'\Subset B\Subset \Omega $, the bounds of that proposition for test measures $(\mathrm{dd}^{\mathrm{c}}w)^ n$, with $w\in \mathrm{PSH}(B)$ and $0\leq w\leq 1$, also sum to $C_{B',B,\Omega }/j$. Dividing by a positive threshold and taking the supremum over $w$ gives convergence in local capacity. These estimates require neither convergence of the targets $\nu _ j$ nor a common local mass bound for them.
Splitting the series defining the metric after its first $j$ terms gives
\[ d_ v\bigl ((\mathrm{dd}^{\mathrm{c}}u_ j)^ n,\nu _ j\bigr )\leq 2/j+2^{-j}\longrightarrow 0. \]
Consequently, the two sequences of measures have the same limits along corresponding subsequences, and hence the same full cluster set $F$.
$\square$
Pome
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