I have problems solving this exercise:
Suppose $A,B \subset \mathbb R^{n}$ are measurable and disjoint. Show that every $E\subset \mathbb R^{n}$ satisfies
$m_{n}^{*}(E \cap (A \cup B))=m_{n}^{*}(E \cap A)+m_{n}^{*}(E \cap B)$
I know that because A and B are disjoint, $E\cap A$ and $E\cap B$ are disjoint.
Also, two sets C and D measurable and disjoint satisfies that $m_{n}^{*}(A \cup B)=m_{n}^{*}( A)+m_{n}^{*}(B)$.
I tried to use the definition of measurability to $A\cup B$ to show that for every $E\subset \mathbb R^{n}$:
$m_{n}^{*}(E)=m_{n}^{*}(E \cap (A \cup B))+m_{n}^{*}(E \setminus (A \cup B))$
Therefore, $m_{n}^{*}(E \cap (A \cup B))=m_{n}^{*}(E \setminus (A \cup B))-m_{n}^{*}(E)$
I don't know how I could continue or what theorem do I need to use. Thank you!