Corollary 13.8.956037F3B58D7

Proof. The inequality $u_{j,k}\leq u_ j$ holds almost everywhere, hence everywhere, since $\max \{ u_{j,k},u_ j\} $ and $u_ j$ are quasi-plurisubharmonic functions that agree almost everywhere. On coordinate charts add smooth local potentials and truncate below at a fixed level $-C$. The resulting locally uniformly bounded plurisubharmonic sequences increase almost everywhere to the truncations of the limits, so they converge in capacity by theorem 7.6.1; by lemma 13.7.3, the original sequences converge in extended capacity. Since $u_{j,k}\leq u_ j$, corollary 13.8.8 bounds every mixed mass of the sequence by that of the limit, and theorem 13.7.5 gives weak convergence and convergence of the masses. This applies in particular to the regularized supremum of an increasing sequence in $\mathrm{PSH}(X,\theta _ j)$ that is bounded above: it belongs to $\mathrm{PSH}(X,\theta _ j)$ by theorem 5.10.2, and it differs from the pointwise supremum only on a pluripolar set, by corollary 5.47.11, hence only on a Lebesgue null set, because plurisubharmonic functions are locally integrable. $\square$