I came to the following question, but I don't have quite a good idea how to approach.


Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position?


I believe that, since this is quite a simple question, this would already have an answer, but I could not find it.



Set of General Linear Position with Nonzero Measure
No, at least if $A$ is assumed to be measurable. Let $\pi \colon \mathbb{R}^n \to \mathbb{R}$ be the projection to the first coordinate. For every $t \in \mathbb{R}$, if $A$ is in general linear position, then $\pi^{-1}(t)$ is finite, and therefore has measure zero in $\mathbb{R}^{n-1}$. So Fubini's Theorem implies that $A$ has measure zero.


