The following properties are equivalent.

• (i)

  Every $\theta$-psh function $\varphi$ is the pointwise limit of a decreasing net of $\theta$-psh model functions.

• (ii)

  For each $u\in C^{0}(X)$ we have

  $$P_{\theta}(u)=\sup\left\{\varphi\mid\varphi\in\mathcal{D}(X)\cap\PSH(X,\omega),\,\varphi\leq u\text{ on }X\right\}.$$

• (iii)

  For each $u\in C^{0}(X)$ $P_{\theta}(u)$ is a uniform limit of $\theta$-psh model functions.