Let $u,u^{\prime}\in C^{0}(X^{\mathrm{an}})$ and $\theta,\theta^{\prime}\in\mathcal{Z}^{1,1}(X)$.

1. (i)

   If $u\leq u^{\prime}$ then ${P}_{\theta}(u)\leq{P}_{\theta}(u^{\prime})$.

2. (ii)

   We have ${P}_{t\theta+(1-t)\theta^{\prime}}(tu+(1-t)u^{\prime})\geq t{P}_{\theta}(u)+(1-t){P}_{\theta^{\prime}}(u^{\prime})$ for all $t\in[0,1]$.

3. (iii)

   We have ${P}_{\theta}(u)+c={P}_{\theta}(u+c)$ for each $c\in\mathbb{R}$.

4. (iv)

   We have ${P}_{\theta}(u)-v={P}_{\theta+dd^{c}v}(u-v)$ for each $v\in{\mathscr{D}}(X)$.

5. (v)

   If ${P}_{\theta}(u)\not\equiv-\infty$, then we have $\sup_{X^{\mathrm{an}}}|{P}_{\theta}(u)-{P}_{\theta}(u^{\prime})|\leq\sup_{X^{\mathrm{an}}}|u-u^{\prime}|$.

6. (vi)

   If $\theta$ is determined on a model ${{\mathscr{X}}}$, if the de Rham class $\{\theta\}$ is ample and if $\theta_{m}\to\theta$ in $N^{1}({{\mathscr{X}}}/S)$, then $P_{\theta_{m}}(u)\to{P}_{\theta}(u)$ uniformly on $X^{\mathrm{an}}$.

7. (vii)

   We have ${P}_{t\theta}(tu)=t{P}_{\theta}(u)$ for all $t\in\mathbb{R}_{>0}$.

8. (viii)

   Assume ${P}_{\theta}(u)\not\equiv-\infty$. Then the envelope $P_{\theta}(u)$ is continuous if and only if it is a uniform limit of $\theta$-psh model functions.