Finite OUTER measure and measurable set Royden's Real Analysis (4th edition), problem #19 (Chapter 2.5):
Let $E$ have a finite OUTER measure. Show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has a finite outer measure and for which $m^{*}(O - E) > m^*(O)-m^*(E)$.
My question is how can a set of finite measure be not measurable? I know that every set of finite positive measure harbors non-measurable subsets, but how could the whole set $E$ be not measurable when it has a finite measure by assumption?
Thanks.
I have righted the above problem. Sorry all!

Consider using a dyadic mesh argument to construct a family of open sets each of which contains $E$, and then take their intersection. 
