Construct an interesting subset of R which is Lebesgue measurable

Question

How to Construct a subset$A$ of $R$ which is Lebesgue measurable and such that $0<4m(int(A))=2m(A)=m(clo(A))<\infty$ where $m$ denotes the Lebesgue measure in $R$?

I can't think of a single place to start.

Answer 1

Let $A = (0, 0.5) \uplus \left( (0.5, 1) \setminus \mathbb Q \right) \uplus \left( \mathbb Q \cap (1, 2) \right)$. Then:

1. $\text{Int } A = (0, 0.5)$, so $\lambda(\text{Int } A) = 0.5$.
2. $2 \, \lambda(A) = 2 \left( \lambda \left(0, 0.5)\right) + \lambda \left( (0.5, 1) \setminus \mathbb Q \right)\right) = 2$
3. $\text{Cl } A = [0, 2]$, so $\lambda(\text{Cl } A) = 2$