Let $A \subset \Bbb R^n$ be measurable and $B \subset \Bbb R^n$ such that $m^*(B)=0$. Show using the definition that $A \cup B$ is a measurable set.
Solution 1:
$$m^*(E \cap (A\cup B)) + m^*(E \setminus (A\cup B))$$ $$\leq [m^{*}(E \cap A)+m^{*}(E \cap B)]+ m^*(E \setminus A)$$ $$=m^{*}(E)$$ since $m^{*}(E \cap B) \leq m^{*}(B)=0$. The reverse inequality always holds.