Sum of Dirac measures is not regular
Take $A=\{0\}\subseteq \Bbb R$. For every open set $G$ containing $A$ there exists an $\varepsilon>0$ such that $]-\varepsilon, \varepsilon[\subseteq G$. This open interval contains infinitely many $\frac 1 n$. Thus $\mu(G)\geq \mu(]-\varepsilon,\varepsilon[) = \infty$. But $\mu(A)=0$. This shows that $\mu$ is not outer regular.