Let $f:X\to[-\infty,+\infty[$ be any function. We define its $\theta$-psh envelope $P_{\theta}(f)$ as follows. If there does not exist any $\varphi\in\PSH(X,\theta)$ such that $\varphi\leq f$ on $X$ then we set $P_{\theta}(f)\equiv-\infty$. Otherwise, we define $P_{\theta}(f)$ as the usc upper envelope of the set of all $\theta$-psh functions $\varphi$ such that $\varphi\leq f$ on $X$, i.e. we set