Give an example of a set with full Lebesgue measure on $[0,1]$ that is also a countable union of nowhere dense sets. I was asked to prove or give a counterexample of the statement.
Assume $P$ is a subset of $[0,1]$ with Lebesgue measure equal to $1$. Then $P$ is not a countable union of nowhere dense sets.
I find that there are nowhere dense sets with positive measure. So I am not able to derive this from subadditivity.

Here is a counterexample:
For any $n$, there is a fat cantor set $C_n$ of measure $> 1 - 1/n$. Each $C_n$ is nowhere dense, since it is closed and has empty interior.
The union $C_1 \cup C_2 \cup \ldots $ has measure one, and is a countable union of nowhere dense sets.
