# Sets measurable in every affine subspace

Take a non-measurable subset $$S\subseteq [-1,1]$$ and subtract $$S\times \{0\}$$ from the unit disk $$B$$ in $$\mathbb{R}^2$$. The set $$X=B\setminus (S\times \{0\})$$ is measurable by 2-D Lebesgue measure because the part on the real line has 2-D measure 0. I don't want this situation.

My request is: is there a name for subsets $$X$$ of $$\mathbb{R}^n$$ such that $$X\cap L$$ is measurable in $$L$$ (according to the dimension of $$L$$) for every affine subspace of $$\mathbb{R}^n$$?

• So you do allow, for example, a non-measurable (according to arc length)subset of a circle, since that meets each line in at most $2$ points ... ? I am not aware of such a name. See also en.wikipedia.org/wiki/Universally_measurable_set 2 days ago
• You might get some ideas here: Anderson, Robert & Zame, William. (2001). Genericity with Infinitely Many Parameters. Advances in Theoretical Economics. 1. 1003-1003. 10.2202/1534-5963.1003. researchgate.net/publication/… 2 days ago