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?



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> For all $S\subseteq X$ we have $\bar{\mu}({\cal P}(S)) =\mu(S)$.
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Lifting a probability measure to the power set
For notational and conceptual simplicity, assume that $X$ and $P(X)$ are disjoint. 


As Bugs Bunny implicitly suggested: 
Let $\mathcal E = \{\{n\}: n \in X \}$ be the set of singletons. This is a subset of $P(X)$. The measure $\bar\mu$ will concentrate on this set. 


For any $\mathcal A \subseteq P(X)$, we will have $\bar \mu(\mathcal A)=\bar \mu(\mathcal A\cap\mathcal E)$; note that $\mathcal A\cap\mathcal E$ is morally the same as a subset of $X$, so $\mu$ will measure it.


Formally, let $S\_{\mathcal A} = \{n\in X: \{n\}\in \mathcal A\}$ be the ``singleton support'' of $\mathcal A$. Define $\bar \mu (\mathcal A) := \mu(S\_{\mathcal A})$. 


