PomeB40742E0DD08Proposition 8.10.4.B40742E0DD08 Suppose $u_ j\downarrow u$ and $v_ j\downarrow v$ are bounded canonical admissible potentials with a common uniform bound. Then
weakly as currents with signed measure coefficients. The same conclusion holds for $v_ j\mu _{u_ j}\to v\mu _ u$.
Proof. Bounded convergence gives $u_ j\to u$ in $L^1$. Their complex Hessians therefore converge in distributions. The preceding coefficient bounds are uniform, so this is weak convergence of each signed coefficient measure against continuous compactly supported tests. The trace measures also converge weakly. Every one of these measures has total variation bounded by $C_ M\operatorname {Cap}_1(E)^{1/2}$ on a Borel set $E$.
Fix $\epsilon >0$. Quasicontinuity supplies an open $G$ with $\operatorname {Cap}_1(G)<\epsilon $ on whose complement $v$ and every $v_ j$ are continuous. Dini's theorem on the compact set $X\setminus G$ gives uniform convergence there. Extend $v|_{X\setminus G}$ to a continuous function $F$ on $X$ with the same uniform bound, by Tietze's theorem. For any one of the signed coefficient measures $\nu _ j\to \nu $ and a fixed continuous compactly supported test $\psi $, subtract $F\nu _ j$ and $F\nu $. The resulting error is at most
The term with $F$ converges by weak measure convergence. First let $j\to \infty $ and then $\epsilon \downarrow 0$. This proves the assertion for the Hessian coefficients and for the positive trace measures. $\square$
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