Proposition 15.9.3EFAF50966FDF

Proof. The $L^1$ limit satisfies $u\geq \varphi -C$, so the additional hypothesis gives $u\sim \varphi $. We repeat the construction of the finite rooftops in the theorem. Every $p_{j,N}$ is still $\theta $-plurisubharmonic and lies between $\varphi -C$ and $C$, which is enough for contact support and for the boundedness of the exponential weights. The resulting inequality is again

The limits $p_ j$ exist because of the common lower bound, and $p_ j\leq u$ follows from $p_ j\leq u_\ell $ for every $\ell \geq j$ and almost-everywhere convergence. Thus $\varphi -C\leq p_ j\leq u$, so all $p_ j$ and their increasing limit $p$ have the same ordinary singularity type as $\varphi $. Monotone convergence in capacity and weighted lower semicontinuity give $\mathrm{MA}^{\mathrm{ext}}_\theta (p)(\{ p<u\} )=0$, and singular domination applies to $(p,u)$ exactly as before, so $p=u$. The upper envelopes of the tails decrease to $u$, and the same squeezing argument gives convergence in capacity. Neither the finite rooftops nor the individual approximants needed to have the singularity type of $\varphi $. $\square$