Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb R^M)=0$.
This is a very strong transition from having sets with a full measure being annihilated into nullity. Is it possible that for some[every?] $x\in(0,1)$ there exists $N=M[G]$ a generic extension of $M$ such that $\mu^N(\mathbb R^M)=x$?
If the answer is negative in its full generality ($M$ is just any model of ZFC) can we add some assumptions for a positive answer? (e.g. $M\models CH$)
This is really just idle curiosity which could not be satisfied via Google, references to a possible answer would be just as welcomed as a complete answer