Proposition 21.12.2911748980D2DF

Proof. By (16.3.7), we have

It follows from proposition 21.12.20, proposition 21.12.21 and proposition 21.12.23 that $P_{\omega }[\varphi ]\in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$. Moreover, by the same propositions, we have

We spell out the last closure. Set

If $x\in \operatorname{Dom}G_{\varphi }$, then $G_{\varphi }(x)-C\leq G_0(x)$ for $C$ large enough, hence $K(x)=G_0(x)$. If $x\notin \operatorname{Dom}G_{\varphi }$, then $K(x)=\infty $. Since $\Delta (\omega ,\varphi )=\overline{\operatorname{Dom}G_{\varphi }}$ and $G_0$ is continuous on $P$, the lower semicontinuous regularization $\operatorname{cl}K$ is equal to $G_0$ on $\Delta (\omega ,\varphi )$ and equal to $\infty $ outside $\Delta (\omega ,\varphi )$. This is exactly (21.12.30). $\square$