Let $\mu$ be a Radon Measure on $X$ and let $\mu_0$ be the semifinite part of $\mu$ ($\mu_0(K) = \sup \{\mu(F), F \subset K , \mu(F) < \infty \}) $
a) Show that $\mu_0$ is inner regular in all Borel sets
b) $\mu_0$ is outer regular in all Borel set with finite measure( $\mu(E) < \infty)$
My try:
Given that $K \subset E$ then we know that $\mu_0(K) \leq \mu_0(E)$. So using the definition of $\sup$, $\sup\{ \mu_0(K): K \subset E \text{ and } K \text{compact} \}\leq \mu_0(E)$ so I get one inequality, but I'm having problems with the other. The same for part b). This is exercise 7.14 from Folland.