Fix $\varphi\in PSH(X,\omega)$ such that $0\leq\varphi\leq 1$. Fix $t>0$ and set $K_{t}=\{x\in X\,/\,\psi(x)<-t\}$. By Chebyshev’s inequality,

$$\int_{K_{t}}\omega_{\varphi}^{n}\leq\int_{X}(-\psi/t)\omega_{\varphi}^{n}\leq\frac{1}{t}\left[\int_{X}(-\psi)\omega^{n}+nVol_{\omega}(X)\right],$$

where the last inequality follows from corollary 2.3. Taking supremum over all $\varphi^{\prime}$s yields the claim. ∎