The first assertion follows from theorem 3.2. Let $\varphi\in PSH(X,\omega)$, $t\in\mathbb{R}$ and set $K_{t}=\{\varphi<-t\}$. Then $\varphi+t\leq 0$ on $K_{t}$ hence $\varphi+t\leq V_{K_{t},\omega}^{*}$. We infer $\sup_{X}\varphi+t\leq\sup_{X}V_{K_{t},\omega}^{*}$ which yields $T_{\omega}(K_{t})\leq\exp(-\sup_{X}\varphi)\exp(-t)$. ∎