Does there exist such a set?

Question

$R$ is a set of real numbers, m is the Lebesgue measure on $R$.

Does exist a nowhere dense subset $A\subseteq R $, such that $m(A)=+\infty$?

I know that if $A$ is of the first category, there exist such a set.

Thanks a lot.

Answer 1

Note that the fat Cantor set is nowhere dense, has measure $1/2$ and is a subset of the unit interval $[0,1]$. If you take the union of the translates of this set by the integers, you will get a nowhere dense set that is of infinite measure.