proof the regularity of the measure in the Riesz representation theorem real and complex analysis , walter rudin , page 48 , theorem 2.17 , the (b)
how did he prove the inner regularity of the measure , please explain 
 A: A lot of reasoning is always omitted in it, so we have to always fill the gap of proofs of Rudin.
For the proof, you have to consider 2 cases.
Given a set $E$ measurable and $\epsilon>0$, 


*

*$\mu(E)<\infty$


In this case, by (a), we can choose a closed set $F$ with the measure $\mu(E-F)<\epsilon$. 
Then since $F$ is $\sigma$-compact and of finite measure, we can choose a compact set $C\subset F$ such that $\mu(F-C)<\epsilon$ 
(Or, you may consider a increasing sequence of compact sets converging to $F$. Note that finite union of compact sets is compact.) 
Thus we have $\mu(E-C)<2\epsilon$ or $\mu(E)<\mu(C)+2\epsilon$. I think the finiteness condition should be used here.


*$\mu(E)=\infty$


I'd like to leave this case as an exercise since it is almost the same proof as above, except that we cannot use $\epsilon$ argument. Instead, you can choose a closed subset $F$ of infinite measure and a increasing sequence of compact sets in $F$ whose measure converges to $\infty$, which anyway implies that $\mu(E)=sup\{\mu(C): C \text{ is compact and }C\subset E \}$
