If $\mathcal{X}$ is an SNC model on which $\theta$ is determined, then it follows from Proposition 7.6 (ii) that the supremum of any $\varphi\in\PSH(X,\theta)$ is attained on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$. This implies the continuity of $\varphi\mapsto\sup_{X}\varphi$.

To prove properness, recall that $\PSH(X,\theta)$ embeds in $\prod_{\mathcal{X}}C^{0}(\Delta_{\mathcal{X}})$. By Tychonoff’s theorem, the compactness of

is therefore equivalent to the compactness in $C^{0}(\Delta_{\mathcal{X}})$ of the closure of the image of $\mathcal{F}_{C}$ in $C^{0}(\Delta_{\mathcal{X}})$, for each SNC model $\mathcal{X}$ on which $\theta$ is determined. But this is a direct consequence of Corollary 7.7 and Ascoli’s theorem.

For the last statement, it is clear that convergence in $\PSH(X,\theta)$ implies pointwise convergence on $X^{\mathrm{qm}}$ which in turn implies pointwise convergence on $X^{\mathrm{div}}$. Now let $(\varphi_{\alpha})_{\alpha\in A}$ be a net of $\theta$-psh functions converging pointwise to $\varphi\in\PSH(X,\theta)$ on $X^{\mathrm{div}}$. Fix any SNC model $\mathcal{X}$ on which $\theta$ is determined. We must show that $\varphi_{\alpha}$ converges uniformly to $\varphi$ on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$. But $X^{\mathrm{div}}\cap\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$ is the image under $\emb_{\mathcal{X}}$ of the rational points in $\Delta_{\mathcal{X}}$ by Corollary 3.13, and is therefore dense in $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$. The uniform convergence on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$ therefore follows from the equicontinuity statement in Corollary 7.7. ∎