Theorem 15.16.7B7C31259154B

Proof. Apply lemma 15.16.5 with $\lambda =1/j$, and write $v_ j=v_{1/j}$ and $c_ j=c_{1/j}$. Extract a subsequence with $v_ j\to v$ in $L^1$ and almost everywhere, and with $c_ j\to c\in [m,C]$. The limit is normalized, $\theta $-plurisubharmonic, and has minimal singularities. The densities $c_ j\mathrm{e}^{v_ j/j}f$ have a common $L^ p$ bound. Hence corollary 15.15.3, applied with the reference $V_\theta $, upgrades the convergence to uniform convergence of differences on $\Omega $ and identifies the weak $L^ p$ limit of these densities with the density of $\mathrm{MA}_\theta (v)$.

At almost every point, $v_ j$ converges to a finite value, so $v_ j/j\to 0$. Since $v_ j\leq 0$ and $c_ j\leq C$, dominated convergence gives $c_ j\mathrm{e}^{v_ j/j}f\to cf$ in $L^ p$. Therefore $\mathrm{MA}_\theta (v)=cf\, \mathrm{d}V$. The uniqueness of the constant follows from corollary 15.13.3. $\square$