Exercise 1.2.26 (Lebesgue outer measure is not additive)

Consider the sets $E$ and $X$ from the proof of Proposition 1.2.18. From the countable sub-additivity and translation invariance we have

We have demonstrated that $1\le m^{\text{∗}}(X)\le 3$ which contradicts to the first case $m^{\text{∗}}(E)=0$. Thus, $m^{\text{∗}}(E)>0$ (maybe small). Pick an $n\in \mathbb{N}$ large enough so that $1∕n<m^{\text{∗}}(E)$. If $m^{\text{∗}}$ were finitely additive, then for a subset $J\subseteq \mathbb{Q}\cap [-1,1]$ such that $\mathit{\#J}=3n$ we would have the result

But ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{q\in J}q+E$ is a subset of $X$ which has outer measure of at most 3. This contradicts monotonicity, and we conclude that $m^{\text{∗}}$ is not (finitely) additive.