What is the relationship between $m^*(A-B)$ and $m^*(A)-m^*(A\cap B)$

Question

What is the relationship between $m^*(A-B)$ and $m^*(A)-m^*(A\cap B)$.

Here $m^*(A)$ means the exterior measaure of $A$.

Answer 1

First of all, in order to make sense the expression "$m^*(A)-m^*(A\cap B)$", we must have that $m^*(A\cap B)<+\infty$. Otherwise, since $m^*(A)\geq m^*(A\cap B) $, it would follow that $m^*(A)=m^*(A\cap B) =+\infty$, so we wouldn't be able to calculate $m^*(A)-m^*(A\cap B)$.

By subadditivity on outer measures, we have that $m^*(A) \leq m^*(A\cap B) + m^*(A\setminus B)$, then since $m^*(A\cap B)<+\infty$, we conclude that $$ m^*(A) - m^*(A\cap B) \leq m^*(A\setminus B). $$

Unfortunately, this is the only relation that we can get in general. However, if we require the set $B$ to be an splitting set of $m^*$, that is, $$ m^*(A) = m^*(A\cap B) + m^*(A\setminus B), \forall A, $$ then we get that $$ m^*(A) - m^*(A\cap B) = m^*(A\setminus B). $$