15.9 Compactness with a common lower barrierD6040BDF0454

Proof. The canonical representative of the limit also lies between $\varphi -C$ and $\varphi +C$. It suffices to show that every subsequence has a further subsequence converging in capacity. We may therefore assume that $I_ j\leq 2^{-j}$ and that $u_ j\to u$ almost everywhere. For $N\geq j$ define

All these envelopes lie between $\varphi -C$ and $\varphi +C$. The contact theorem and ordered contact give an inequality for the finite rooftops. Partition the contact set of $p_{j,N}$ according to the first index $\ell $ for which $p_{j,N}=u_\ell $. On the corresponding piece, the measure of $p_{j,N}$ is bounded by that of $u_\ell $, and $\mathrm{e}^{p_{j,N}}=\mathrm{e}^{u_\ell }$. Therefore

Only positive measures enter this argument, so it applies also when their total masses are infinite.

As $N\to \infty $, $p_{j,N}$ decreases to $p_ j$; the limit is the envelope of the infimum because every competitor lies below every finite rooftop. The sequence $p_ j$ increases, and its canonical limit $p$ is $\theta $-plurisubharmonic. By lemma 5.29.19, both monotone limits hold in capacity. Applying lemma 13.14.8 to the bounded nonnegative weights $|\mathrm{e}^{p_{j,N}}-\mathrm{e}^ u|$, first in $N$ and then in $j$, we obtain

The inequalities $p_ j\leq u_\ell $ for all $\ell \geq j$ imply $p\leq u$, first almost everywhere and then for the canonical representatives. Also $p\sim u\sim \varphi $. Off the pole sets, $\mathrm{e}^ p<\mathrm{e}^ u$ exactly on $\{ p<u\} $, so $\mathrm{MA}^{\mathrm{ext}}_\theta (p)(\{ p<u\} )=0$. Since $P_{\delta \theta }(\varphi )-C$ is a competitor for $P_{\delta \theta }(p)$, the latter is not identically $-\infty $. Theorem 15.2.2, applied with coefficient zero, gives $p\geq u$, and hence $p=u$.

Finally, by Hartogs' lemma and the almost-everywhere convergence, $U_ j=(\sup _{\ell \geq j}u_\ell )^*$ decreases to $u$. The inequalities $p_ j\leq u_ j\leq U_ j$, together with the convergence in capacity of both monotone sequences, prove the claim. $\square$

Proof. The $L^1$ limit satisfies $u\geq \varphi -C$, so the additional hypothesis gives $u\sim \varphi $. We repeat the construction of the finite rooftops in the theorem. Every $p_{j,N}$ is still $\theta $-plurisubharmonic and lies between $\varphi -C$ and $C$, which is enough for contact support and for the boundedness of the exponential weights. The resulting inequality is again

The limits $p_ j$ exist because of the common lower bound, and $p_ j\leq u$ follows from $p_ j\leq u_\ell $ for every $\ell \geq j$ and almost-everywhere convergence. Thus $\varphi -C\leq p_ j\leq u$, so all $p_ j$ and their increasing limit $p$ have the same ordinary singularity type as $\varphi $. Monotone convergence in capacity and weighted lower semicontinuity give $\mathrm{MA}^{\mathrm{ext}}_\theta (p)(\{ p<u\} )=0$, and singular domination applies to $(p,u)$ exactly as before, so $p=u$. The upper envelopes of the tails decrease to $u$, and the same squeezing argument gives convergence in capacity. Neither the finite rooftops nor the individual approximants needed to have the singularity type of $\varphi $. $\square$

Proof. In the first case, lemma 5.44.2, applied to the exponentials $\mathrm{e}^{u_ j}$, proves (15.9.2). In the second case we use lemma 5.44.7 instead, since these exponentials are bounded and have a common lower bound for their curvature. In both cases the criterion gives convergence in capacity.

The masses are uniformly bounded by those of the dominating measures. For the deep sublevels we use $u_ j\geq \varphi -C$, which gives $\{ u_ j<-T\} \subset \{ \varphi <C-T\} $. In the first case the masses of these sublevels are at most $\nu (\{ \varphi <C-T\} )$, which tends to zero as $T\to \infty $ because $\nu $ does not charge the pole set of $\varphi $. In the second case lemma 5.29.16 bounds them by $C'\operatorname {Cap}_\omega (\{ \varphi <C-T\} )$, which also tends to zero. The tail criterion theorem 13.15.13, in the general-background form of theorem 13.15.23, now gives finiteness of the limiting measure and weak convergence. $\square$