On page 33 of Stable Mappings and Their Singularities by Golubitsky and Guillemin (GTM 014; 1974), the following proposition is characterized as an "obvious, but surprisingly complicated result":
Proposition 1.10: Let S be a nonempty open subset of $\mathbb{R}^n$. Then S is not of measure zero.Edit: As pointed out in the comments, Gowers already mentioned an equivalent result whose difficulty is more clear. Please vote this answer down (I can't vote down my own answers).