Union of non measurable set

Question

Can I have an example of two disjoint sets A,B where $\mu^*(A \cup B) =\mu^*(A)+\mu^*(B) $ is not always right?

Answer 1

There exists $A\subset [0,1]$ such that whenever $C$ is an uncountable closed subset of $[0,1]$ we have $$A\cap C \ne \phi \ne ([0,1] \backslash A)\cap C.$$

Let $B=[0,1]$ \ $A.$ Let $\mu^o$ and $\mu^i$ denote, respectively, outer and inner Lebesgue measure.

Any closed subset of $A$ or of $B$ is countable, so $\mu^i(A)=0=\mu^i(B).$ We have $\mu^o(A)=1-\mu^i(B)=1$ and $\mu^o(B)=1-\mu^i(A)=1.$

So $\mu^o(A)+\mu^o(B)=2$, but $\mu^o(A\cup B)=\mu^o([0,1])=1.$