Let $X\neq\emptyset$ be a set and let $\mu:{\cal P}(X)\to [0,1]$ be a probability measure. Is there a probability measure $$\bar{\mu}:{\cal P}({\cal P}(X))\to [0,1]$$ with the following property?

For all $S\subseteq X$ we have $\bar{\mu}({\cal P}(S)) =\mu(S)$.