Suppose $A \subset T$ and $B \cap T = \emptyset$. Show that $m^*(A \cup B) = m*(A)+m*(B)$.
My thoughts: I know that $m^*(A \cup B) \leq m^*(A)+m^*(B)$ by having already proven this, so it will be enough to show that $m^*(A \cup B) \geq m^*(A)+m^*(B)$. I know and understand the Caratheodory Condition, and am pretty sure I need to use it, but I'm not sure what test set to consider. I'm working through this myself and would most appreciate a hint rather than a full proof. Thanks!