Let $r\in[0,1]$. We look at the binary represenation of $r$ and say that $r$ is *binarily universal* if every finite binary string appears in at least one place in the binary representation of $r$. Let $U$ be the set of binarily universal members of $[0,1]$.


Is $U$ a Borel set? If yes, what is its Borel measure?



Binarily universal members of $[0,1]$
$U$ is a $G\_\delta $ set, hence Borel at level 2 of the Borel hierarchy. It has Lebesgue measure 1 and is also comeager, so it is large both in the sense of measure and of Baire category. 


