If $E \subseteq \mathbb{R}^n$ and $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is an orthogonal map, then I know that $\mu^*(QE)=\mu^*(E)$, i.e., the Lebesgue outer measure is invariant under rotations.
Now I would like to show that $E$ is measurable iff $QE$ is measurable, using the Carathéodore criterion.
Do you have any idea?
Thanks in advance.