Corollary 15.15.3131F17206F6A

Proof. Applying the theorem in both orders to $u_ j$ and $u_ k$, with the common reference $\eta $, we see that an $L^1$-Cauchy sequence in the family is uniformly Cauchy in differences on $\Omega $. Its uniform limit on $\Omega $ is locally $\theta $-plurisubharmonic and agrees almost everywhere, hence everywhere for the canonical representatives, with the $L^1$ limit $u$. The bounds $\eta -C_0\leq u\leq \eta $ hold on all of $X$. Letting $k\to \infty $ in the pairwise estimate proves the quantitative bound, before any equation for $u$ is used.

The densities form a bounded set in the reflexive space $L^ p$. Let $f$ be the weak limit of a subsequence $f_{u_{j_ k}}$. Since the potentials are locally bounded and converge locally uniformly on $\Omega $, Bedford–Taylor continuity, which holds also for the smooth nonclosed background, gives $\mathrm{MA}_\theta (u)=f\, \mathrm{d}V$ on $\Omega $. Neither measure charges the analytic set $Z$, so this equality holds on $X$. Positivity and weak lower semicontinuity of the norm give $f\geq 0$ and $\| f\| _ p\leq L$. Thus every weak cluster point equals the density $f_ u$ of $\mathrm{MA}_\theta (u)$, which proves weak convergence of the whole sequence. Finally, the common bounds by $\eta $ and $\eta -C_0$ give $L^1$ compactness of the family, by the compactness of quasi-plurisubharmonic functions, and the preceding argument upgrades each convergent subsequence to uniform convergence of differences on $\Omega $. $\square$