Proposition 13.21.3EE58E71478F6

Proof. Normalize $u_ j\leq 0$ by a common translation. Set $p_ j=P_{\theta _ j}[u_ j]$, $p=P_\theta [u]$ and $q=\lim _ jp_ j$. Since $\underline{\operatorname {vol}}_{\theta _ j}(u_ j)\geq \underline{\operatorname {vol}}_\theta (u)>0$, theorem 13.19.5 in background $\theta _ j$ shows that every $p_ j$ is an ordinary model with the same volumes as $u_ j$. The envelopes decrease, and the uniform convergence of the backgrounds gives $q\in \mathrm{PSH}(X,\theta )$. Since $u\leq p\leq q$, the limit $q$ has positive lower volume, and theorem 13.21.2 gives, for either volume,

If $q=p$, these equalities and theorem 13.19.5 show that both volumes of $u_ j$ converge to those of $u$; in particular the upper volumes do. Conversely, if the upper volumes converge, then $u$ and $q$ have the same upper volume, so $u\in \mathcal E_{\mathrm{vol}}(X,\theta ,q)$. By corollary 13.19.8, $P_\theta [u]=q$, that is, $q=p$. For the Lelong assertion, proposition 13.20.6 shows that the upper-volume deficit between $u$ and $u_ j$ in background $\theta _ j$ tends to zero. Apply proposition 13.18.3 with the common lower bound $\underline{\operatorname {vol}}_\theta (u)>0$ and the uniform Lelong bound for $\pi ^*\theta _1$. $\square$