Let $\mathcal{X}^{\prime}\geq\mathcal{X}$ be two SNC models on which $\theta$ is determined. Then $p_{\mathcal{X}}\circ p_{\mathcal{X}^{\prime}}=p_{\mathcal{X}}$. By Proposition 7.5 (ii) this implies $\varphi\leq\varphi\circ p_{\mathcal{X}^{\prime}}\leq\varphi\circ p_{\mathcal{X}}\circ p_{\mathcal{X}^{\prime}}=\varphi\circ p_{\mathcal{X}}$, with equality on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$. Set $\tilde{\varphi}:=\lim_{\mathcal{X}}\varphi\circ p_{\mathcal{X}}$. Then $\tilde{\varphi}\geq\varphi$. On the other hand, it follows from Corollary 3.2 that $p_{\mathcal{X}}$ converges to the identity on $X$, so by upper semicontinuity of $\varphi$ we have $\varphi\geq\tilde{\varphi}$. ∎

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. ∎

Condition (i) implies that $u\circ p_{\mathcal{X}}=u\circ i_{\mathcal{X}}\circ p_{\mathcal{X}}$ is continuous for all $\mathcal{X}$, so that $v:=\inf_{\mathcal{X}}u\circ p_{\mathcal{X}}$ is usc. It follows that $v\geq u^{*}$, since $v\geq u$ by (ii). Conversely, for each $x\in X$ we have $\lim_{\mathcal{X}}p\mathcal{X}(x)=x$, hence

which shows that $v=u^{*}$. Finally, (ii) shows that $\mathcal{X}^{\prime}\geq\mathcal{X}\Rightarrow u\circ p_{\mathcal{X}^{\prime}}\leq u\circ p_{\mathcal{X}}$, hence $v\circ p_{\mathcal{X}}=u\circ p_{\mathcal{X}}$, which is equivalent to the last assertion. ∎

Upon considering the new family $\varphi_{I}=\max_{\alpha\in I}\varphi_{\alpha}$ with $I$ ranging over all finite subsets of $A$, we may assume that $A$ is a directed set and $(\varphi_{\alpha})$ is an increasing net. For each SNC model $\mathcal{X}$ we have $\varphi_{\alpha}\leq\varphi_{\alpha}\circ p_{\mathcal{X}}$ for all $\alpha$, hence $\varphi\leq\varphi\circ p_{\mathcal{X}}$. By Corollary 7.7 $\varphi_{\alpha}\circ i\mathcal{X}$ converges uniformly to $\varphi\circ i_{\mathcal{X}}$, which is therefore continuous. Using Lemma 7.10 we conclude that $\varphi^{*}$ is usc, satisfies $\varphi^{*}\leq\varphi^{*}\circ p_{\mathcal{X}}$, and $\varphi^{*}\circ i_{\mathcal{X}}=\varphi\circ i_{\mathcal{X}}$ is a uniform limit of restrictions to $\Delta_{\mathcal{X}}$ of $\theta$-psh functions, hence $\varphi^{*}$ is $\theta$-psh. ∎