Let $\theta$ be as above. A $\theta$-psh function $\varphi:X\to[-\infty,+\infty[$ is an usc function such that for each SNC model $\mathcal{X}$ of $X$ on which $\theta$ is determined we have

1. (i)

   $\varphi\leq\varphi\circ p_{\mathcal{X}}$ on $X$;

2. (ii)

   the restriction $\varphi\circ\emb_{\mathcal{X}}$ of $\varphi$ to $\Delta_{\mathcal{X}}$ is a uniform limit of restrictions of $\theta$-psh model functions.

We write $\PSH(X,\theta)$ for the set of $\theta$-psh functions on $X$.