Given a set $A_q \subseteq R$,$n \in Z$ ,$q \in Q$ ,$A_n=A_q+nq$, $\bigcup_n A_n=R$,{$A_n$} pairwise disjoint for a given $q$.
$A_q+nq$ means set $A_q$ has all it's elements translated by an amount $n*q$.
is $A_{q}$ necessary a measurable set?
It is not difficult to see if $ R$ is replaced with open interval $(a, b) $ then $ A_q $ must be of the form $ A_q = I \bigcup S $ where $ I $ is a countable union of intervals and $ S $ is a countable set.