Proposition 18.6.68B21A468DD06

Proof. Choose finitely supported probability measures $\nu _ j$ with rational weights converging weakly to $\theta $, and let $q_ j$ be their normalized Green potentials; the preceding proof gives $q_ j\to 0$ in $L^1$. Put $a=1-\tau $ and $r_ j=P_\theta [a q_ j]_{\mathcal I}$. The scaled potential lies in the same background, since

Since $aq_ j$ is a competitor, $aq_ j\leq r_ j\leq 0$, and hence $r_ j\to 0$ in $L^1$. By projection, $r_ j$ is an ideal model. By proposition 18.4.8, it has exactly the logarithmic coefficients of $aq_ j$, which are rational, with bounded local remainders, and it is locally bounded away from the finite pole set. On the curve, removing the polar part of the curvature therefore gives

By distributional Stokes, this measure has total mass $1-a=\tau $. The positive curvature measures $\theta +\mathrm{dd}^{\mathrm{c}}r_ j$ have mass one and converge as distributions, hence weakly, to $\theta $. Since $\nu _ j\rightharpoonup \theta $, the last identity gives $\mathrm{MA}_\theta (r_ j)\rightharpoonup \tau \theta $. Finally, $P_\theta [0]_{\mathcal I}=0$, so the distance in the statement is $\int _ Y|r_ j|\, \mathrm{d}V$, which tends to zero. The limiting potential $0$ has non-pluripolar measure $\theta $, of mass one. $\square$