Theorem 17.5.13109ED281DA22

Proof. By corollary 17.4.5, the set $K_\phi $ of potentials $w\in \mathrm{PSH}(X,\theta )$ with $w\leq \phi $ and $\sup _ Xw=0$ is compact in $L^1(X)$; it contains $\phi $. By corollary 17.5.12, the function $w\mapsto \int _ X\mathrm e^{\lambda w}\, \mathrm d\nu $ is continuous on $K_\phi $. Its values are positive, since the pole set of $w$ is $\nu $-null and $\nu (X)>0$. Hence it has a positive minimum $a$ on $K_\phi $.

Any $v\in \mathcal E^1(X,\theta ,\phi )$ can be written $v=w+c$ with $c=\sup _ Xv$ and $w\in K_\phi $, by lemma 16.22.2. Since $E(w)\leq 0$, the shift formula for the energy gives

The right side tends to $-\infty $ as $c\to \pm \infty $. Since $F_\lambda (\phi )$ is finite, a maximizing sequence $v_ j=w_ j+c_ j$ with $F_\lambda (v_ j)\geq F_\lambda (\phi )-1$ therefore has bounded constants $c_ j$. Moreover $E(v_ j)\geq F_\lambda (v_ j)$, because the exponential term is nonnegative, so $E(w_ j)=E(v_ j)-mc_ j$ is bounded below. By corollary 17.4.5, after passing to a subsequence, $c_ j$ converges and $w_ j$ converges in $L^1(X)$; hence $v_ j\to u$ in $L^1(X)$ for some $u\in \mathcal E^1(X,\theta ,\phi )$. The energy is upper semicontinuous (proposition 17.4.4), and $\int _ X\mathrm e^{\lambda v_ j}\, \mathrm d\nu \to \int _ X\mathrm e^{\lambda u}\, \mathrm d\nu $ by corollary 17.5.12, so $F_\lambda (u)\geq \varlimsup _ jF_\lambda (v_ j)=\sup F_\lambda $, and $u$ is a maximizer. By lemma 17.5.9, $\mu _ u=\mathrm e^{\lambda u}\nu $. Uniqueness in $\mathcal E(X,\theta ,\phi )$ is proposition 17.5.7. $\square$