Suppose that $X$ is a compact space. Let $M(X)=C(X)^*$ denote the Banach space of regular measures. Is the following statement true:

$F:M(X)\to\mathbb{C}$ is a positive functional if and only if the corresponding element in $C(\widetilde{X})$ is a positive function, where $\widetilde{X}$ denotes the hyper-stonean cover of $X$. (see for example mathoverflow post for)