There is a set called Vitali Set which is not Lebesgue measurable. Analogously, there also exists a Vitali set $Y$ in $\mathbb R^3$ which is a subset of $[0,1]^3$ and $|Y\cap q|=1$ for all $q\in ...
(Before I start, I don't know which tag is suitable for this post. Please retag my post if it bothers you.) Let's say there is a string on $[0,1]$ with a mass given by $m(x)$. ($m(x)$ means the mass ...