Can a set of positive measure and its complement both have empty interior?

Question

This might be silly, but I am not sure:

Does there exist a Lebesgue measurable subset $E \subseteq (0,1)$ such that

1. $E$ and $(0,1) \setminus E$ both have positive Lebesgue measure.

2. $E$ and $(0,1) \setminus E$ both have empty interiors.

If we relax condition $1$, then $E=Q\cap (0,1)$ works. If we relax condition $2$, then the fat Cantor set does the job. (Its complement have non-empty interior though).

Answer 1

Let $E$ be the union of $(0,1) \setminus \mathbb Q)\cap (0,\frac 1 2]$ and $\mathbb Q\cap (\frac 1 2,1)$. Then $E$ and $(0,1)\setminus E$ both have positive measure and they have no interior.

Answer 2

Let $E$ be a fat Cantor set $C$ together with the $C^\complement\cap\mathbb Q$.