TITLE: Lebesgue measure of sets of empty interior
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: Fat Cantor sets provide another set of counter examples.