This is a question about forcing. I have seen the following fact mentioned in multiple places, but have not been able to find a proof: if a random real is added to a transitive model of ZFC, then in the generic extension the set of reals in the ground model becomes meager. 


My guess is that one should be able to, in some natural way, directly construct from a random real a countable sequence of nowhere dense sets covering the ground model reals, but I am not sure.



Adding a random real makes the set of ground model reals meager
The proof is based on the fact that there is a decomposition ${\bf R}=A\cup B$ of the reals such that $A$, $B$ are (very simple) Borel sets, $A$ is meager, $B$ is of measure zero, and ${\bf R}=A\cup B$ even holds if after forcing we reinterpret the sets. Nos let $s$ be a random real. If $r\in {\bf R}$ is an old real, then $s\notin r+B$, so $s\in r+A$, that is, the meager 
$s-A$ contains all old reals. 


