Let $\mu$ be a probability measure on $X$ which satisfies condition ${\mathcal{H}}(\alpha,A,\omega)$. Then there exists a unique continuous function $\varphi\in PSH(X,\omega)$ such that

$$\mu=(\omega+dd^{c}\varphi)^{n}\;\text{ and }\;\sup_{X}\varphi=-1.$$

Moreover $||\varphi||_{L^{\infty}(X)}\leq C$, where $C$ only depends on $\alpha,A$ and $\omega$.