Lebesgue measure of sets of empty interior

Question

There is a proposition to the effect that if the Lebesgue measure of a set is zero, then it is a set of empty interior. Does the converse also hold true, i.e. is the Lebesgue measure of any set of empty interior zero? If not, is there a straightforward counterexample?

Answer 1

What about $\mathbb{R}\setminus\mathbb{Q}$? It has empty interior but it has infinite measure because $\mathbb{Q}$ has measure zero (it is countable).

Answer 2

Fat Cantor sets provide another set of counter examples.