Can $\mathbb R$ be partitioned into two sets with positive measure on every interval?

Question

An answer in this post got me wondering: Suppose $A \subseteq \mathbb R$ is Borel measurable, and let $B = \mathbb R \setminus A$. Is it possible that, for every nonempty open interval $I \subseteq \mathbb R$, we could have $\lambda(A \cap I) > 0$ and $\lambda(B \cap I) > 0$, where $\lambda$ is the Lebesgue measure? Please provide an example or proof of impossibility.