Question 18 of Section 2.5 (page 43) in this link : http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf claims that if a set $ E $ has finite outer measure, then, there is a set $ F \in F_{ \delta} $ such that, we have $ F \subset E $ and $ m^{*}(F) = m ^ { * } (E) $.
However, I am able to prove that I can find such an $ F $ only if $ E $ is measurable. Can someone figure out if there is any mistake in the proof below? Or is the exercise an oversight?
Proof. Suppose that there is an $ F \in F_{ \delta } $ such that $ m ^{ * } ( F) = m ^{ * } ( E ) $. Then, we apply the excision property (page 40 of Royden) on the measurable set $ F $, which is of finite measure, to deduce that $ m ^ { * } ( E - F ) = m ^ { * } ( E ) - m ^{*} (F) = 0 $. Then, by Theorem 11 (iv) (page 40), we deduce that $ E $ is measurable.
