Exterior measure on $\mathbb{Q}$

Question

I'm preparing myself for the final exam in real-analysis and I'm trying to solve this exercise.
Show that there is no exterior measure $\mu^*$ on $\mathbb{Q}$ s.t. $\forall$ $p, q$ $\in$ $\mathbb{Q}$ with $p<q$ $$ \mu^*({x \in \mathbb{Q} : p\leq x\leq q})=q-p.$$
I was thinking to suppose that there is such an exterior measure, and then I'd probably get a contradiction, but I don't know how it should be done...

Answer 1

$\mu^{*}(\{a\})=0$ for each $a$ and $\mathbb Q$ is countable so $\mu^{*}(A)=0$ for every $A$ by countable sub-additivity of exterior measures.