Let $\mu$ be a probability measure and $t>0$. Let $\varphi,\psi\in PSH(X,\omega)$ be such that $\int_{X}e^{t\varphi}d\mu=\int_{X}e^{t\psi}d\mu=1$. Assume $\varphi\in{\mathcal{E}}^{1}(X,\omega)$ is a global solution to the complex Monge-Ampère equation $(\omega+dd^{c}\varphi)^{n}=e^{t\varphi}\mu$, while $\psi\in{\mathcal{C}}^{0}(X\setminus E)$ satisfies $(\omega+dd^{c}\psi)^{n}=e^{t\psi}\mu$ only in $X\setminus E$.

Then $\psi\in{\mathcal{E}}^{1}(X,\omega)$ and $\psi\equiv\varphi$.