Lemma 5.25.145B3FA75B7BD

Proof. On a coordinate ball about $x$, add a common smooth correction $q$ so that $u_ j+q$ and $u+q$ are plurisubharmonic. For a fixed small radius, the submean inequality bounds $(u_ j+q)(x_ j)$ by the average of $u_ j+q$ over the ball of that radius centered at $x_ j$. As $j\to \infty $, these averages converge to the average of $u+q$ over the ball of the same radius centered at $x$: the contribution of $u_ j-u$ tends to zero in $L^1$, and the integral of the fixed locally integrable function $u+q$ over a ball depends continuously on its center. Letting the radius decrease to zero proves the inequality.

The same estimate gives local uniform upper bounds for the $u_ j$ with $j$ large. By coefficient freezing and the local upper-envelope theorem, the tail envelopes are therefore quasi-plurisubharmonic, with the same lower Hessian bound. They decrease with $j$. If their limit exceeded $u(x)$ at some point $x$, the definition of the regularization would give indices $k_ j\geq j$ and points $x_ j\to x$ contradicting the first assertion for the subsequence $(u_{k_ j})$. On the other hand, an almost everywhere convergent subsequence of $(u_ j)$ shows that $h_ j\geq u$ almost everywhere, hence everywhere by lemma 5.24.3. Thus $h_ j\downarrow u$. $\square$