Proposition 17.3.1369D86F9D2F90

Proof. First let the net be a sequence $(u_ j)$. By monotonicity $E_\theta ^\phi (u_ j)$ decreases to a limit $L\geq E_\theta ^\phi (u)$. Let $w\in \mathcal E^\infty (X,\theta ,\phi )$ with $w\geq u$, and put $v_ j=\max \{ u_ j,w\} $. Then $v_ j$ decreases pointwise to $\max \{ u,w\} =w$. Since $u_ j\leq u_1\leq \phi +C$ and $w\geq \phi -C$ for some constant $C$, the differences $v_ j-w$ are uniformly bounded. The tangent bound (17.3.10) gives

by bounded convergence. Since $E_\theta ^\phi (u_ j)\leq E_\theta ^\phi (v_ j)$, we get $L\leq E_\theta ^\phi (w)$, and taking the infimum over $w$ gives $L\leq E_\theta ^\phi (u)$.

For a general net, corollary 10.3.6 shows that $u_ i\to u$ in $L^1(X)$; its hypothesis holds because $\sup _ Xu_ i\geq \sup _ Xu>-\infty $. Choose indices $i_1\leq i_2\leq \cdots $ with $\| u_{i_ k}-u\| _{L^1}<1/k$. The decreasing pointwise limit $\tilde u$ of $(u_{i_ k})$ is a $\theta $-psh function by theorem 5.10.1, applied in charts after adding smooth local potentials of $\theta $, since it dominates $u$ and so is not identically $-\infty $. By monotone convergence, $\int _ X(\tilde u-u)\, \omega ^ n=\lim _ k\int _ X(u_{i_ k}-u)\, \omega ^ n=0$, so $\tilde u=u$ almost everywhere, hence everywhere. The sequence case gives $E_\theta ^\phi (u_{i_ k})\to E_\theta ^\phi (u)$. The net $E_\theta ^\phi (u_ i)$ is decreasing, so its limit is its infimum, which is at most $\lim _ kE_\theta ^\phi (u_{i_ k})=E_\theta ^\phi (u)$; the reverse inequality is monotonicity. (The selected indices need not be cofinal in $I$; this is why we compare infima.)

For concavity let $u,v\preceq \phi $ and $0<t<1$; we may assume that $E_\theta ^\phi (u)$ and $E_\theta ^\phi (v)$ are finite, since otherwise there is nothing to prove. The potentials $(1-t)\max \{ u,\phi -k\} +t\max \{ v,\phi -k\} $ belong to $\mathcal E^\infty (X,\theta ,\phi )$ and decrease to $(1-t)u+tv$ as $k\to \infty $. Concavity on $\mathcal E^\infty (X,\theta ,\phi )$ and decreasing continuity give

In particular the finite locus of the energy is convex. $\square$