This problem comes from page 99 in Folland's "real analysis: modern techniques and their applications", 2nd edition, as the image below shows.
I tried to prove condition (ii) for n=1 under condition (i), The author says it follows from Theorems 1.16 and 1.18, but I can not see any relation between condition (ii) and the two Theorems mentioned. What $\inf\{...\}$ defines is an outer measure. But, since the collection of all open set, i.e., the topology of $\mathbb R^n$, is not a ring, I can not prove that the outer measure restricted to $\mathbb R^n$ is a measure. In consequence, I can not use subtractivity property of measure to prove condition (ii) using the method similar to that of Theorem 1.14. Could you please help me with this problem? Thanks!
PS: Since this is a specific problem in Folland's book, in addition to the good will to answer, I have to assume the answerer have at hand a copy of Folland's book and, as a must have for owners of this book, its two versions of erratas.