(i) The only thing to show is that $P_{\theta}(u)$ is $\theta$-psh. Since $P_{\theta}(u)\leq u$ and $u$ is continuous, it follows that the usc regularization satisfies $P_{\theta}(u)^{*}\leq u$. Now, $P_{\theta}(u)^{*}$ is $\theta$-psh by Theorem 7.9, and is hence a competitor in the definition of $P_{\theta}(u)$. Thus $P_{\theta}(u)=P_{\theta}(u)^{*}$ is indeed $\theta$-psh.

(ii) is trivial.

(iii) follows from the fact that given $\varphi\in\PSH(X,\theta)$, $\varphi^{\prime}\in\PSH(X,\theta^{\prime})$ with $\varphi\leq u$ and $\varphi^{\prime}\leq u^{\prime}$, $t\varphi+(1-t)\varphi^{\prime}$ belongs to $\PSH(X,t\theta+(1-t)\theta^{\prime})$ and is dominated by $tu+(1-t)u^{\prime}$.

(iv) and (v) are seen similarly.

(vi) is a formal consequence of (ii) and (iv).

(vii) By Proposition 5.2 we may assume after perhaps passing to a higher model that there exists a model function $v$ determined on $\mathcal{X}$ such that $\theta+dd^{c}v$ is $\mathcal{X}$-positive, i.e. determined by an ample class in $N^{1}(\mathcal{X}/S)$. As a consequence, there exists an open neighborhood $V\subset N^{1}(\mathcal{X}/S)$ of $\theta$ such that $\tau+dd^{c}v$ is $\mathcal{X}$-positive for all $\tau\in V$.

We claim that $P_{\tau}(u)$ is uniformly bounded on $X$ for $\tau\in V$. Indeed for each $\tau\in V$ we have $\mathbf{R}\subset\PSH(X,\tau+dd^{c}v)$, hence $P_{\tau+dd^{c}v}(u-v)\geq\inf_{X}u-\sup_{X}|v|$. By (v) it follows that

which proves the claim.

Now for each $x\in X$ the function $\tau\mapsto P_{\tau}(u)(x)$ is concave on $V$, hence locally Lipschitz continuous on $V$, with local Lipschitz constant only depending on $\sup_{\tau\in V}|P_{\tau}(u)(x)|$, which is in turn bounded independently of $x\in X$, and the result follows. ∎