15.37.12 Common envelopes and dominated continuityBD25F1249EB6

Proof. We first bound stretched rooftops of the tail of the sequence from below and then add the finitely many remaining terms.

The suprema of an $L^1$-convergent sequence are bounded, so (15.37.2) gives a constant $C$ with $u_ j\leq \varphi +C$ for all $j$. After a common translation we may assume $u_ j\leq \varphi \leq 0$, and then $u\leq \varphi $. Since $\mu _{u_ j}\leq \nu $, lemma 5.44.2 gives

We extract, once and for all, a subsequence with $I_ j\leq 2^{-j}$ along which $u_ j\to u$ almost everywhere for smooth volume.

For $N\geq j$ put $p_{j,N}=P_\theta (u_ j,\ldots ,u_ N)$. These rooftops belong to $\mathcal E_\varphi $ by proposition 16.28.64. Partition their contact sets according to the first index at which the minimum is attained. Ordered contact on these disjoint pieces gives both

The first bound has no factor $N$, because each piece is controlled by the restriction of the same measure $\nu $ to that piece.

Fix a real number $B\geq 1$ and set $r_{j,N}=Q_ B(p_{j,N},\varphi )$. It belongs to $\mathcal E_\varphi $ by the stretching lemma, so $\mu _{r_{j,N}}(X)\geq \kappa $. With $D_{j,N}=\{ B^{-1}r_{j,N}+(1-B^{-1})\varphi =p_{j,N}\} $, contact support and positivity give

On this contact set, $\varphi \leq 0$ gives $\mathrm{e}^{p_{j,N}}\leq \mathrm{e}^{r_{j,N}/B}$. Consequently

On the other hand, choose $K>0$ with $B^ n\nu (\{ u\leq -K\} )<\kappa /2$; this is possible because $u\not\equiv -\infty $ and $\nu $ does not charge its pole set. The measure domination then gives $\mu _{r_{j,N}}(\{ u\leq -K\} )<\kappa /2$, and the lower mass bound yields

Choose $j$ so large that $B^ n2^{1-j}<\mathrm{e}^{-K}\kappa /4$. Then (15.37.15) bounds $\sup _ Xr_{j,N}$ from below, independently of $N$. Since the sequence decreases in $N$ and is bounded above by $\varphi $, its limit $R$ is not identically $-\infty $. It satisfies, for every $\ell \geq j$,

The second inequality follows already from $r_{j,N}\leq p_{j,N}\leq \varphi $.

It remains to include the finitely many earlier terms and to verify every defining condition of the relative class. Fix $b\geq 1$, apply the preceding construction with $B=2b$, and let $j$ and $R$ be as above. If $j>1$, let $p_0=P_\theta (u_1,\ldots ,u_{j-1})$ and $S=Q_{2b}(p_0,\varphi )$. By the finite-rooftop and stretching lemmas, neither is identically $-\infty $, and $S\leq \varphi $. If $j=1$, set $S=\varphi $. The potential $H=(R+S)/2$ is $\theta $-plurisubharmonic. For a tail index $\ell \geq j$ we use (15.37.16) and $S\leq \varphi $; for an earlier index we use the analogous inequality for $S$ and $R\leq \varphi $. In either case,

For $b=1$ this shows that $h=P_\theta (\inf _\ell u_\ell )$ is not identically $-\infty $. For general $b$ the left side is a competitor for $h$, so the same inequality holds with $h$ in place of every $u_\ell $; this is the defining condition of the relative class for $b$. Also $h\leq \varphi $. Thus $h\in \mathcal E_\varphi $.

Finally $h\leq u_ j$ for all $j$, so the almost-everywhere convergence and lemma 5.24.3 give $h\leq u\leq \varphi $. Upward stability proves $u\in \mathcal E_\varphi $. Undoing the common translation proves the lemma as stated. $\square$

Proof. The bounds pass to $u$ almost everywhere along a subsequence, and then everywhere by lemma 5.24.3; upward stability gives $u\in \mathcal E_\varphi $. After translating the entire family and its bounds by a common constant, we may assume that all potentials are nonpositive.

It suffices to prove convergence in capacity along a further subsequence of every subsequence. Extract so that $I_ j\leq 2^{-j}$ and $u_ j\to u$ almost everywhere for smooth volume, and form

All these envelopes lie above $h$ and below $\varphi +C$; hence none of them is identically $-\infty $, and all belong to $\mathcal E_\varphi $. Partition the contact set of $p_{j,N}$ according to the first index $\ell $ with $p_{j,N}=u_\ell $. Ordered contact on each part gives

Only integrals of nonnegative functions occur here, so the estimate holds even if a rooftop has infinite mass.

As $N\to \infty $, $p_{j,N}\downarrow p_ j$; as $j\to \infty $, the $p_ j$ increase, and their regularized limit $p$ is not identically $-\infty $. Both monotone limits hold in capacity by lemma 5.29.19. Moreover $h\leq p\leq u\leq \varphi +C$, where $p\leq u$ follows from the almost-everywhere convergence of the extracted $u_ j$. Thus $p\in \mathcal E_\varphi $.

Apply lemma 13.14.8, first as $N\to \infty $ and then as $j\to \infty $, with the bounded quasi-continuous weights in the preceding estimate. These weights converge in capacity because the exponential is Lipschitz on a common upper half-line. We obtain

The second equality follows off the poles, which the extended measures ignore. The zero-domination lemma lemma 15.37.9 now gives $p=u$; no finiteness of $\mu _ p$ is needed.

Finally, the upper envelopes $U_ j=(\sup _{\ell \geq j}u_\ell )^*$ decrease to $u$, by theorem 5.25.2 and the almost-everywhere convergence. Applying lemma 5.29.19 to $p_ j\uparrow u$ and $U_ j\downarrow u$, the inequalities $p_ j\leq u_ j\leq U_ j$ give convergence in capacity along this subsequence, and hence for the original sequence. $\square$