I am quite sure the following fact must have been known for set theorists, though I could not find it anywhere.


If $r$ is random over $L$ and $x\in L[r]\setminus L$, then there must be some real $r\_0$ random over $L$ so that $r\_0\in L[x]$ and $x\in L[r\_0]$.


So where I could find it?


**Addition**


Monroe Eskew has given an answer to the question. I would like to explain the phenomenon from a recursion theory point of view.


As Jason pointed out, the result is corresponded to Demuth's theorem in recursion theory. Actually it can be proved by applying Demuth's argument.


Suppose that $r$ is random and $x\in L[r]\setminus L$. Then there is a condition $r\in p\Vdash x\mbox{ is a real}$. Since random forcing is c.c.c., we may in $L$ find a sequence $\{p^i\_n\}\_{i,n\in \omega}$ stronger than $p$ so that for every $n$, $\{p^i\_n\}\_{i\in \omega}$ is a maximal antichain below $p$ and each $p^i\_n$ decides a value of $x$. Coding these condition and the relation $p^i\_n\Vdash x(n)=j\_i$ into a single real $z\in L$. Then $x\leq\_T r\_0\oplus z$ with a use function $f\in L[r]$. Then $f$ is dominated by a function $g\in L$. Now using these facts and applying Demuth's argument, we may obtain a real $r\_0$ which is also $L$-random so that $x\oplus z\oplus g\equiv\_T r\_0\oplus z\oplus g$.


So $x\in L[r\_0]$ and $r\_0\in L[x]$.



Demuth's theorem in set theory
I'm not sure if this specific claim is stated explicitly anywhere, but it follows from the more general discussion about intermediate extensions in Jech, page 247. If we take any $x \in L[r]$, then there is a complete subalgebra $A$ of the random algebra associated to $\dot{x}$, with the property that $L[x]=L[G\_r \cap A]$. But any complete subalgebra of a measure algebra is also a measure algebra. So $A$ is isomorphic to random forcing. 


