Outer Measure of the complement of a Vitali Set in [0,1] equal to 1
Hint: By Theorem 3.2, $m_*(I)=m_*(U)+m_*(U^c).$
Recall the key property of the Vitali set $N$ is that the sets $N \oplus q_i$, where $\oplus$ is addition mod 1 and $\mathbb{Q} = \{q_1, q_2, \dots\}$, are disjoint. (Not sure if this is the same notation that S&S use.)
Now, consider the sets $U^c \oplus q_i$. Show that they are disjoint measurable sets and each has the same measure as $U^c$, which is greater than $\epsilon$. Therefore, what can you say about the measure of $V = \bigcup_i U^c \oplus q_i$? This should give you a contradiction.