Let's view $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and pick some basis $(v_\alpha)_{0 \leq \alpha < \mathfrak{c}}$ of it. We can then consider the subspace $L$ spanned by $(v_\alpha)_{0 < \alpha < \mathfrak{c}}$ (ie leaving out one vector from the basis). Given the horrible way we have built $L$, I don't suppose there is much a priory reason for $L$ to be measurable. However, I am wondering whether we can say something about the outer measure of $L$?