TITLE: Can anyone give an example of a closed set contains no interval but with finite non-zero Lebesgue measure?
QUESTION [5 upvotes]: Can anyone give an example of a closed set $F$ of $\Bbb{R}$ such that $0<|F|<+\infty$ and $F$ contains no open interval? Thank you!

REPLY [6 votes]: Because $\mathbb {Q}$ has measure $0,$ there is an open set $U$ containing $\mathbb {Q}$ such that $m(U) < 1/2.$ Then $[0,1]\setminus U$ is closed (in fact compact), and contains no interval (because it contains no rational). We have $m([0,1]\setminus U) > 1/2,$ so we're done.