Good point. According to Bogachev, a Borel measure $\mu$ is called - inner regular if $\mu(B)=sup\{\mu(K): K\subset B, K \mbox{is closed}\}$ for any Borel $B$, and - tight if $\mu(B)=sup\{\mu(K): K\subset B, K \mbox{is compact}\}$ for any Borel $B$.

Could you possibly recommend any good book/survey article on the subject?

"To a man with a hammer, everything looks like a nail." Mark Twain