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 $q\in \mathbb R^3/\mathbb Q^3$. However, I'm curious about if it fulfilled a kind of isotropic uniform medium, let this isotropic uniform medium has density $\rho$, and put it on a electronic scale to weigh, what reading can we get? Note that $m_Y=\rho V_Y$ but $V_Y$ seems to be undefined... So it seems we cannot get any real reading. But on the other hand, since we are using a electronic scale, it also seems we must get a reading...A paradox?