TITLE: How to prove the difference of two regular complex Borel measures is regular?
QUESTION [7 upvotes]: This is on page 130 of Rudin's real & complex analysis. The book says that this is easy to see, but I'm having trouble proving it. 
Suppose $\mu_1,\mu_2$ are two regular complex Borel measures. For inner regularity, given $E$ and $\epsilon >0$, we have $K_1,K_2\subset E$ such that $|\mu_1|(K_1)+\epsilon >|\mu_1|(E)$ and $|\mu_2|(K_2)+\epsilon >|\mu_2|(E)$. I want to use $K_1\cup K_2$ for $|\mu_1 + \mu_2|$ (consider sum of the measures instead of difference), but it does not seem to work.
Is this technique on the right track?

REPLY [5 votes]: (1) Show that if $\mu$ is a regular complex Borel measure, then so is $-\mu$.
(2) Now use the fact that the sum of two regular complex Borel measures is a regular complex Borel measure (which you know how to prove).
I hope this helps!