Let $O$ be a fixed point of the plane $\mathbb R^2$. A set $S$ of points in the plane has the property that for any real number $r > 0$, at most one point of $S$ lies on the circle centered at $O$ with radius $r$.
I am hoping that this might be enough to show that $S$ is Lebesgue measurable with measure zero. Is this true?
(This question came up in a discussion with a student of mine, so I don't know if it's easily answerable.)