outer measure problem when 2 sets are positively separated In this question if we assume that two sets are not positively separated but intersection is zero will there be a contradiction.
My try: First of all the set should not be measurable. But if it is not i do not know how to associate an outermeasure.If anyone can help it would be great. Thanks
If I understand correctly, you ask whether $ \mu^*(A\bigcup B)=\mu^*(A)+\mu^*(B)$ holds when $A$ and $B$ are disjoint. It does not.
Take a Vitali set $V$. Since a line segment can be covered by countably many sets $V+q_k$, the subadditivity of $\sigma$ implies $\mu^*(V)>0$. On the other hand, the disjoint union of $V+q_k$ is contained in a line segment of length $k$. Take an integer $n$ such that $n\mu^*(V)>3$. Then $$\mu^*\left( \bigcup_{k=1}^n (V+q_k)\right)\le 3 < \sum_{k=1}^n \mu^* (V+q_k)$$ which means that additivity fails for finite disjoint unions.