Take a non-measurable subset $S\subseteq [-1,1]$ and subtract $S\times \{0\}$ from the unit ball $B$ 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 $\mathbb{R}^n$?

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 $\mathbb{R}^n$?