[BFJ11, Proposition 7.4]. The set $\PSH(X,\theta)$ is convex. If $\varphi,\psi$ are $\theta$-psh and $c\in\mathbf{R}$, then the functions $\max\{\varphi,\psi\}$ and $\varphi+c$ are also $\theta$-psh.