$\text Carothers' Real Analysis$ defines his Vitali cover with no introduction which made me confused a lot.
Here is the definition of a Vitali Cover:
I'm not sure when does a set have its Vitali Cover? It seems any subset of $\mathbb R$ does exist its Vitali Cover. If it is true, how about $E$ be the set of all rationals in $\mathbb R$(or say rational set $\mathbb Q$)? What kind of Vitali Cover does $\mathbb Q$ have?
P.S. Symbol "$m$" denote Lebesgue Measure that is Lebesgue Outer Measure under Lebesgue measurable set.