When two sets are positively separated we know that $\mu(A \cup B)=\mu(A)+\mu(B)$. My question is what happens when their intersection is null. Will the above equation be invalid?
My Try:It has to be that the sets A or B should be non measurable or else if both are measurable then i can show that the above equation is valid. But when considering outer measures for non measurable sets i am having a lot of problems. I do not know how to assign an outer measure to non measurable sets. So if anyone can help it would be great. Thank you.
Let's use Lebesgue measure in the real line. There is a (non-measurable) subset $A$ of $[0,1]$ such that both $A$ and its complement $B=[0,1]\setminus A$ have outer measure 1. Then, of course $A \cap B = \varnothing$, but $\mu(A)+\mu(B) = 2 \ne 1 = \mu([0,1]) = \mu(A \cup B)$.