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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 $\mathbb{R}^n$?

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  • $\begingroup$ 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 $\endgroup$ 2 days ago
  • $\begingroup$ 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/… $\endgroup$
    – John Levy
    2 days ago

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