PomeA72166484AB2Let $\Omega \subset \mathbb C^ n$ be a domain, $n\geq 1$, and let $b$ be a locally bounded plurisubharmonic function on $\Omega $. If potentials approach $b$ from below in local capacity, every limit of their Monge–Ampère measures must dominate $(\mathrm{dd}^{\mathrm{c}}b)^ n$. Truncation at a fixed distance below $b$ provides a common bound, and plurifine locality keeps the measure outside the small set where the truncation changes the potential.
Lemma 5.27.41.CD407359DDEA Let $b,u_ j$ be locally bounded plurisubharmonic functions on $\Omega $ with $u_ j\leq b$ and $u_ j\to b$ in local capacity. Then, for every nonnegative $f\in C_ c(\Omega )$,
In particular, every vague subsequential limit of $(\mathrm{dd}^{\mathrm{c}}u_ j)^ n$ dominates $(\mathrm{dd}^{\mathrm{c}}b)^ n$.
Proof. First suppose that $f$ is supported in $B'$, where $B'\Subset B\Subset \Omega $ are concentric balls. Put $v_ j=\max (u_ j,b-1)$ and $E_ j=\{ u_ j\leq b-1\} $. The functions $v_ j$ lie between $b-1$ and $b$ and tend to $b$ in local capacity. Because of their common bounds, the continuity of bounded products recalled in section 5.27 gives $(\mathrm{dd}^{\mathrm{c}}v_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}b)^ n$ vaguely on $B$. With $R=\sup _ B b-\inf _ B b+1$, the capacity bound for bounded potentials gives
where the limit holds because $E_ j\subset \{ |u_ j-b|>1/2\} $. On the plurifine open Borel set $\{ u_ j>b-1\} $ we have $v_ j=u_ j$, so theorem 5.27.1 shows that the two measures agree there. By positivity,
Taking the lower limit proves the assertion for such $f$. For a general $f$, use a finite nonnegative partition of unity subordinate to a cover of its support by such balls. Applying the inequality along a vaguely convergent subsequence proves the final assertion. $\square$
Theorem 5.27.42.745A94C03209 Assume that $\Omega $ is bounded, $n\geq 2$, and $b\in \mathrm{PSH}(\Omega )\cap L^\infty (\Omega )$. For a set $F\subset \mathcal M_+(\Omega )$, the following are equivalent:
$F$ is vaguely closed and every $\mu \in F$ satisfies $\mu \geq (\mathrm{dd}^{\mathrm{c}}b)^ n$;
there are bounded plurisubharmonic functions $u_ j\leq b$ on $\Omega $, converging to $b$ in $L^ p(\Omega )$ for every $1\leq p<\infty $ and in local capacity, whose full vague cluster set is $F$.
The empty set is allowed. If $b$ is smooth, every $u_ j$ in the second condition can be chosen smooth.
Proof. If the second condition holds, lemma 5.27.41 gives the measure inequality, and the full cluster set is closed in the metrizable vague topology by the argument at the beginning of the proof of theorem 5.27.39.
Conversely, assume the first condition. If $F$ is nonempty, the concatenation of a dense sequence used in the proof of theorem 5.27.39 gives measures $\nu _ j\in F$ whose full vague cluster set is $F$. If $F$ is empty, take $\nu _ j=(\mathrm{dd}^{\mathrm{c}}b)^ n+j\delta _ a$ for a fixed $a\in \Omega $. A nonnegative compactly supported test function equal to one at $a$ has integral at least $j$ against $\nu _ j$, so this sequence has no vaguely convergent subsequence. In either case, $\nu _ j-(\mathrm{dd}^{\mathrm{c}}b)^ n$ is a positive Radon measure.
Let $(f_ i)$ and $d_ v$ be as in lemma 5.27.38. For each $j$, approximate the excess $\nu _ j-(\mathrm{dd}^{\mathrm{c}}b)^ n$ against $f_1,\ldots ,f_ j$ by a positive finite atomic measure, with each error less than $1/j$. If this measure has $N_ j>0$ atoms, perform the construction from the proof of theorem 5.27.36, starting at $b$ and using $\tau =t=(jN_ j)^{-1}$ for each atom; if it has none, take $u_ j=b$. Summing the estimates of proposition 5.27.35 gives
For every fixed pair of concentric balls $B'\Subset B\Subset \Omega $, the same construction bounds the integral of $|u_ j-b|$ over $B'$ against $(\mathrm{dd}^{\mathrm{c}}w)^ n$ by $C_{B',B,\Omega }/j$, for every $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Dividing by a fixed positive threshold and taking the supremum over $w$ gives convergence in local capacity. Each $u_ j$ is bounded and lies below $b$, and it is smooth when $b$ is smooth. One sequence serves all finite $p$, and no common local mass bound on the $\nu _ j$ is needed.
The error in the first $j$ coordinates and the tail of the series yield
By the remark following lemma 5.27.38, the two sequences of measures have the same limits along corresponding subsequences. Hence their full vague cluster sets coincide. $\square$
For example, on the unit ball the function $b=|z|^2-1$ has nonzero Monge–Ampère measure. For every point $a$ of the ball, the theorem allows the cluster sets $\{ (\mathrm{dd}^{\mathrm{c}}b)^ n\} $, $\{ (\mathrm{dd}^{\mathrm{c}}b)^ n+\delta _ a\} $ and the empty set, but it excludes $\{ 0\} $; this exclusion relies on the convergence in local capacity. In dimension one, $L^1$ convergence alone forces $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}b$ vaguely, so the full cluster set is always $\{ \mathrm{dd}^{\mathrm{c}}b\} $.
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