PomeCD407359DDEALemma 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. $\square$
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