My professor asked this question:
Let $E \subseteq \mathbb{S}^{n-1}$ be a Borel set and let $E_1 = \{x\in \mathbb{R}^n \setminus \{0\}:|x|\in (0,1], \frac{x}{|x|} \in E\}.$
Show that $\sigma(E) = nm(E_1)$ is a Borel measure on $\mathbb{S}^{n-1}$.
I'm super confused about how to approach this and what the proof would look like - any help is appreciated.