Intuition about open cover of $\mathbb Q$ of arbitrary small measure I have a question that is a bit vague. One can construct (more ore less explicitly) an open cover of say $\mathbb Q\cap[0,1]$ of arbitrary small Lebesgue-measure. My question asks for an explanation of how one should think about such a cover. The best would be a sort of visualisation (but this is certainly asking for too much) but my imagination is puzzled by the fact that an arbitrary big set (in a measure-theoretic sense) of [0,1] can be left uncovered while the rationals are covered and they are dense... Does anyone has a explanation of this?

I don't think a sane persion would easily visualize such a construct. The construct will be a dense set with just isolated points cut out (since $\mathbb Q$ is dense).
The construct is to simply enumerate the rational numbers and surround them with an open set such that $m(V_n) = \epsilon k^{-n}$. Since $\bigcup V_n$ will cover all rational numbers and $m\bigcup V_n \le \sum \epsilon k^{-n} = \epsilon / (1-k)$ you can thereby cover the $\mathbb Q$ with arbitrarily small open set.
