Theorem 17.5.15373141BD8F91

Proof. Let $\lambda _ j\downarrow 0$, let $u_ j$ be the solution of $\mu _{u_ j}=\mathrm e^{\lambda _ ju_ j}\nu $ given by theorem 17.5.13, and write $u_ j=w_ j+c_ j$ with $c_ j=\sup _ Xu_ j$. Since $u_ j\preceq \phi $ and $\sup _ Xw_ j=0$, lemma 16.22.2 gives $w_ j\leq \phi $, so $w_ j$ belongs to the set $K_\phi $ of corollary 17.4.5, which is compact in $L^1(X)$. On $K_\phi $ the functions

are continuous, by corollary 17.5.12, and they increase pointwise to the constant $m$, by monotone convergence, since $w\leq 0$ and the pole set of $w$ is $\nu $-null. The convergence is uniform on $K_\phi $ (Dini's theorem): for $\varepsilon >0$ the compact sets $\{ w\in K_\phi :m-H_ j(w)\geq \varepsilon \} $ decrease and have empty intersection, so they are empty for large $j$.

Comparing total masses in the equation gives $m=\mathrm e^{\lambda _ jc_ j}H_ j(w_ j)$, so

Also $\int _ X|1-\mathrm e^{\lambda _ jw_ j}|\, \mathrm d\nu =m-H_ j(w_ j)\to 0$. Hence $r_ j=\mathrm e^{\lambda _ ju_ j}=\mathrm e^{\lambda _ jc_ j}\mathrm e^{\lambda _ jw_ j}\to 1$ in $L^1(\nu )$, and $\mu _{w_ j}=\mu _{u_ j}=r_ j\nu $. Pass to a subsequence along which $r_ j\to 1$ $\nu $-almost everywhere and, by compactness of $K_\phi $, $w_ j\to u$ in $L^1(X)$ for some $u\in K_\phi $. Lemma 17.5.14 gives $\mu _ u\geq \nu $. Since $u\leq \phi $, mass monotonicity gives $\mu _ u(X)\leq m=\nu (X)$. Hence $\mu _ u=\nu $, $u$ has full mass relative to $\phi $, and $\sup _ Xu=0$ because $u\in K_\phi $. $\square$