closed subset that has the same size

Question

Suppose that $\psi(A)>0$ where $\psi$ is some measure and $A$ is an open set. Is it true that we can always find a closed set such that $B \subseteq A$ and $\psi(B)=\psi(A)$?

I have a follow up question here (what if I change the question to $\psi(B)>0$ instead of $\psi(B)=\psi(A)$): Existence of closed subset of positive measure

Answer 1

No. Suppose that $\psi$ is the Lebesgue measure on $\mathbb R$ and that $A=(0,1)$. If $B$ is a closed subset of $A$, then $\psi(B)<1=\psi(A)$.

Answer 2

No. The reason is that the complement of a closed subset is open and that open sets have a non-zero measure in most of the usual cases.