Define $\lambda^*:\wp\left(\mathbb R\right)\to\left[0,\infty\right]$ by
$$\lambda^*\left(E\right)=\inf\left\{\sum_{n\in\mathbb N}\left(b_n-a_n\right):E\subseteq\bigcup_{n\in\mathbb N}\left(a_n,b_n\right)\right\}.\tag{$\star$}$$
In other words, $\lambda^*$ is the Lebesgue outer measure.
Clearly, $\lambda^*\left(\left\{q\right\}\right)=0$ for every $q\in\mathbb R$. Therefore, since outer measures are countably subadditive, $\lambda^*\left(\mathbb Q\right)=0$;
$$\lambda^*\left(\mathbb Q\right)=\lambda^*\left(\bigcup_{q\in\mathbb Q}\left\{q\right\}\right)\leq\sum_{q\in\mathbb Q}\lambda^*\left(\left\{q\right\}\right)=0.$$
However, how does this follow from $\left(\star\right)$ alone? I am having a hard time coming up with a sequence of collections of intervals whose unions contain $\mathbb Q$ and whose sums approach zero since $\mathbb Q$ is dense in $\mathbb R$.