I wonder if there is a construction for a closed subset of $[0,1]$ such that it has no interior (contains no open intervals) and has measure exactly $1$ as $[0,1]$ itself.
I know the fat Cantor set, but I think fat Cantor set can be extended to construct any closed subsets of $[0,1]$ with measure $a <1$. I don't think it can be extended to exactly $1$. However, I also cannot find the contradiction that if there is such a set. Any suggestions ?