I have a question which asks if the following is true: if $A,B\subseteq \mathbb{R}$ are closed, disjoint sets that are unbounded, then $m^*(A\cup B)=m^*(A)+m^*(B)$. I think it is true due to the fact that $A$,$B$ are closed which implies they are Borel (so measurable). But then I don't see how the unbounded bit comes into the question. Am I missing a trick here? thank you.
edit: $m^*$ is the outer lebesgue measure.
