By Theorem A’, we can choose $\varphi$ be a continuous $\omega$-psh function satisfying $\MA(\varphi)=\mu$. Set $t_{i}=\varphi(x_{i})$ for $i=1,...,N$. We claim that $\varphi_{S,t}=\varphi$, which will conclude the proof. On the one hand we have $\varphi\leq\varphi_{S,t}$ by (8.4), since $\varphi$ is $\omega$-psh and satisfies $\varphi(x_{i})\leq t_{i}$. On the other hand we have $\varphi_{S,t}=\varphi$ on the support of $\MA(\varphi)$, hence $\varphi_{S,t}\leq\varphi$ by Lemma 8.4. ∎