Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $0 < \lambda(A\cap(a,b))< (b-a)$

Question

Question: Is there a Lebesgue measurable subset $A \subset R$ such that for every interval $(a,b)$ we have $$0 < \lambda(A\cap(a,b))< (b-a)$$

It looks like the answer is no, I'm trying to use the Lebesgue differentiation theorem on A in order to do so.

Also, If we assume there's a set $A$, $\forall (a,b) \subset (0,1)$, $0 < \lambda(A\cap(a,b))< (b-a)$, Does there exist an open interval $I_n \subset (0,1)$ that will hold as a contradiction?