Proposition 21.12.21419A1D9F18A9

Proof. By proposition 16.3.2, we have $\varphi \land \psi \in \mathrm{PSH}(X,\omega )$. Since $\varphi $ and $\psi $ are $T_ c$-invariant, the set of candidates in the definition of $\varphi \land \psi $ is preserved by the action of $T_ c$. Hence $\varphi \land \psi $ is $T_ c$-invariant as well. So $\varphi \land \psi $ is the biggest element in $\mathrm{PSH}_{\mathrm{tor}}(X,\omega )$ which is dominated by both $\varphi $ and $\psi $. In view of theorem 21.12.13 and proposition 21.12.19, $G_{\varphi \land \psi }$ is the smallest closed proper convex function $G$ on $M_{\mathbb {R}}$ dominating both $G_{\varphi }$ and $G_{\psi }$, which is just $G_{\varphi }\lor G_{\psi }$.

For the statement about $F$, we use proposition 21.11.36:

Since $\varphi \land \psi \not\equiv -\infty $, $F_{\varphi \land \psi }=(G_\varphi \lor G_\psi )^*$ is finite on all of $N_{\mathbb R}$. It follows that $F_\varphi \land F_\psi >-\infty $, and hence is finite on all of $N_{\mathbb R}$ as well. Being finite and convex on $N_{\mathbb R}$, it is continuous, hence closed. Therefore