Theorem 11.33.55E3583383688

Proof. Choose solutions $u_ j\sim \varphi $ normalized by $\sup _ Xu_ j=0$, and write $c_ j=c_\varphi (F_ j)$. If $|F_ j|\leq M$, monotonicity and the translation rule of proposition 11.33.2 give

In particular, the Monge–Ampère measures of the $u_ j$ are dominated by a fixed multiple of $\mathrm{d}V$, and theorem 15.12.3 gives a constant $C$ such that

From any subsequence, extract a further subsequence with $c_ j\to c>0$ and $u_ j\to u$ in $L^1(X)$. By corollary 15.9.4, the convergence also holds in capacity, $u\sim \varphi $, and the measures converge weakly. On the other hand, the exponential function is Lipschitz on the common range of the $F_ j$, so $c_ j\mathrm{e}^{F_ j}\to c\mathrm{e}^ F$ in $L^1(\mathrm{d}V)$. Therefore

and uniqueness of the constant gives $c=c_\varphi (F)$. Since every subsequence has a further subsequence along which $c_ j\to c_\varphi (F)$, the whole sequence converges, and so do the logarithms. $\square$