When $d(A,B)>0$, we have $$m^*(A)+m^*(B)=m^*(A\cup B)$$I want to examine whether the identity holds if we only have $A\cap B=\varnothing$.
My attempt:
The proof of the origin theorem requires a $\delta$-distance to separate these 2 sets. So the only worse situation may only happen if they have strange boundaries. However, I am unable to construct a counterexample so far.