Let $X$ be a locally compact metric space which is $\sigma$-compact, and let $\mu$ is an unsigned Borel measure which is finite on every compact set. Show that $\mu$ is a Radon measure.
I know that every unsigned Borel measure on a compact metric space which is $\sigma$-compact is a Radon measure. From the assumption, we only need to verify outer regularity and inner regularity. Inner regularity is easier, since we can write $X$ as a countable union and compacts sets $K_n$, and a closed set in each $K_n$ is also a closed set in $X$. I have difficulty verifying outer regularity, open set in $K_n$ may not be open in $X$.