Pick $s,t>0$. Since (6.1) holds for bounded $\omega$-psh functions, we see using (3.1) that

$$-\infty<E\left(\frac{\varphi+\psi}{2}\right)\leq E\left(\frac{\varphi^{\langle t\rangle}+\psi^{\langle s\rangle}}{2}\right)\leq\frac{2^{-(n+1)}}{n+1}\int\psi^{\langle s\rangle}\MA(\varphi^{\langle t\rangle}).$$

Since $\psi^{\langle s\rangle}$ decreases to $\psi$ at any point of $X$, the right hand side converges to

$$\frac{2^{-(n+1)}}{n+1}\int\psi\MA(\varphi^{\langle t\rangle})\leq\frac{2^{-(n+1)}}{n+1}\int_{\{\varphi>-t\}}\psi\MA(\varphi^{\langle t\rangle})=\frac{2^{-(n+1)}}{n+1}\int_{\{\varphi>-t\}}\psi\MA(\varphi)$$

by monotone convergence. We obtain the desired estimate by letting $t\to\infty$. ∎