Theorem 13.19.5CB29AE1DFDA3

Proof. Since $u\preceq p$, monotonicity gives $\underline{\operatorname {vol}}_\theta (u)\leq \underline{\operatorname {vol}}_\theta (p)$ and $\overline{\operatorname {vol}}_\theta (u)\leq \overline{\operatorname {vol}}_\theta (p)$. In particular, the lower volumes agree if $\underline{\operatorname {vol}}_\theta (p)=0$. Suppose henceforth that $\underline{\operatorname {vol}}_\theta (p)>0$. Normalize $u\leq 0$, so that $u\leq p$, and choose any $w\in \mathrm{PSH}(X,\theta )$ with $w\sim p$. Fix $K\geq 0$ with $p-K\leq w\leq p+K$. The rooftop envelope $p_ C=P_\theta (u+C,w)$ satisfies $u-K\leq p_ C\leq u+C$ for each $C\geq 0$, so it is $\theta $-plurisubharmonic and equivalent to $u$. Its regularized increasing limit $r=P_\theta [u](w)$ satisfies

Indeed, the leftmost potential lies below $u+C$, and below $p-K\leq w$ because $P_\theta (u+C+K,0)\leq P_\theta [u]=p$; it is therefore a competitor for the rooftop. Letting $C\to \infty $ gives the lower bound for $r$. Thus $r\sim p$, and proposition 16.28.62 puts $r$ in $\mathcal E_{\mathrm{env}}(X,\theta ,p)$.

By lemma 13.19.4, $\mu _ r$ vanishes on $\{ r<w\} $. Theorem 13.19.1, with coefficient $c=0$ and reference $p$, gives $r\geq w$; hence $r=w$. Since $p_ C\to r$ in capacity, lower semicontinuity of the total mass (lemma 13.14.8) and $p_ C\sim u$ yield

Taking the supremum over $w\sim p$ gives $\overline{\operatorname {vol}}_\theta (p)\leq \overline{\operatorname {vol}}_\theta (u)$, and hence equality of the upper volumes. Theorem 13.18.5 then gives $u\in \mathcal E_{\mathrm{env}}(X,\theta ,p)$ and equality of the lower volumes. Finally, proposition 16.28.65 gives $P_\theta [p]=P_\theta [u]=p$. $\square$