Theorem 14.19.3 (Nguyen)CFA579B45DF9

of theorem 14.19.3. Choose smooth strictly positive functions $f_ j\to f$ in $L^ p$. By theorem 14.7.2, there are smooth $u_ j$ with $\omega _{u_ j}>0$ and $\omega _{u_ j}^ n=\mathrm{e}^{\lambda u_ j}f_ j\, \mathrm{d}V$. The preceding lemma bounds $M_ j=\sup _ Xu_ j$, and consequently bounds $F_ j=\mathrm{e}^{\lambda u_ j}f_ j$ in $L^ p$. The normalized potentials $\psi _ j=u_ j-M_ j$ are uniformly bounded by theorem 14.19.19. We may pass to a subsequence with $M_ j\to M$ and $\psi _ j$ Cauchy in $L^1$. Theorem 14.19.21 gives uniform convergence $u_ j\to u\in \mathrm{PSH}(X,\omega )\cap C(X)$.

Uniform convergence and $f_ j\to f$ in $L^ p$ give $\mathrm{e}^{\lambda u_ j}f_ j\to \mathrm{e}^{\lambda u}f$ in $L^ p$, and continuity of bounded products therefore gives $\omega _ u^ n=\mathrm{e}^{\lambda u}f\, \mathrm{d}V$. Uniqueness, even among all bounded solutions, follows from bounded exponential comparison, corollary 14.19.12. $\square$