Let $X \subset \mathbb R$ be a Lebesgue measurable set, with $\mu(X)>0$. Then for every $\alpha \in (0, \mu(X))$ there exists a measurable subset $X_\alpha \subset X$ such that $\mu(X_\alpha) = \alpha$.
Could you please provide any hints, please? I've thought about inner/outer regularity of Lebesgue measure but I can't manage to finish. For example, we know that for every $\varepsilon>0$ there exists a close subset $C \subset X$ s.t. $\mu(X \setminus C)<\varepsilon$... but how can I use this?
Thanks in advance.