My question is very short: does there exist a couple of sets $A,B\subset[0;1]$ such that $A\cap B=\emptyset$, but $\mu(A)=\mu(B)=1$? Here $\mu(\cdot)$ is outer measure.
It's easy to construct disjoint $A,B$ such that $\mu(A)+\mu(B)\ne\mu(A\cup B)$ as a kind of Vitali set, but such an example doesn't answer my question. Of course, $A$ and $B$ have to be not measurable.