Can anyone give an example of a closed set contains no interval but with finite non-zero Lebesgue measure? 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!

Enumerate the rational points of $[0,1]$ as a sequence $(r_n)_{n\in\mathbb N}$. Then, choose a sequence of positive numbers $(\varepsilon_n)$ such that $\sum_1^\infty 2\varepsilon_n<1$ and set $K:=[0,1]\setminus \bigcup_1^\infty ]r_n-\varepsilon_n, r_n+\varepsilon_n[$. This is a closed set with finite measure; it contains no non-trivial interval because the sequence $(r_n)$ is dense in $[0,1]$, and $\vert K\vert$ is at least $1-\sum_1^\infty 2\varepsilon_n>0$.
