Let $S=[0,1]\setminus \mathbb{Q}$ and let $A\subset S$ be a closed subset with a positive Lebesgue measure ($=\mathcal{L}(A)>0$).
I want to show the following.
Let $b\in[0, \mathcal{L}(A))$ then there exists a closed subset $B\subset A$ such that $\mathcal{L}(B)=b$.
This is just my assumption and I am not sure if the statement holds. Hence, I would appreciate any hints that would help me prove/disapprove this statement.