Proposition 21.13.580728D8B152D3

Proof. By (16.3.7),

Hence $P_{\theta }[\varphi ]\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ by proposition 21.13.55, proposition 21.13.56, proposition 21.13.57. The same results give

Put

If $x\in \operatorname{Dom}G_{\varphi }$, then $G_{\varphi }(x)-C\leq G_{V_{\theta }}(x)$ for all large enough $C$, so $K(x)=G_{V_{\theta }}(x)$. If $x\notin \operatorname{Dom}G_{\varphi }$, then $K(x)=\infty $. By proposition 21.11.34, we have $\Delta (\theta ,\varphi )=\overline{\operatorname{Dom}G_{\varphi }}$. Moreover, theorem 21.13.45 gives $F_{V_{\theta }}\sim \operatorname{Supp}_{P_ D}$, hence $G_{V_{\theta }}\sim \chi _{P_ D}$ by Legendre duality. In particular, $G_{V_{\theta }}$ is finite and continuous on $P_ D$. Thus the lower semicontinuous regularization of $K$ is equal to $G_{V_{\theta }}$ on $\Delta (\theta ,\varphi )$ and equal to $\infty $ outside $\Delta (\theta ,\varphi )$. $\square$