Set $\varphi_{j}=\lambda^{-j}\varphi\circ f^{j}$. Observe that $\varphi_{j}$ is uniformly bounded from above and that $\varphi_{j}+g_{j}\in PSH(\mathbb{C}\mathbb{P}^{n},\omega)$. It follows from proposition 1.6 that either $\varphi_{j}$ converges uniformly towards $-\infty$ or it is relatively compact in $L^{1}(\mathbb{C}\mathbb{P}^{n})$. It is sufficient to show that for $A>0$ large enough, $\overline{\lim}_{j\rightarrow+\infty}T_{\omega}(\varphi_{j}<-A)<T_{\omega}(X)=1$. Observe that $f^{j}(\varphi_{j}<-A)=\{\varphi<-A\lambda^{j}\}$. Therefore

$$\left[\alpha T_{\omega}(\varphi_{j}<-A)\right]^{\lambda^{j}}\leq T_{\omega}(\varphi<-A\lambda^{j})\leq C\exp(-A\lambda^{j}),$$

where the last inequality follows from proposition 3.8. We infer

$$\overline{\lim}_{j\rightarrow+\infty}T_{\omega}(\varphi_{j}<-A)\leq\frac{1}{\alpha}\exp(-A)<1$$

for $A>-\log\alpha$ large enough. ∎