Fix $r>0$ and define $S_r=\{z \in \mathbb{R}^3: |z|=r\}$. My question is: which subsets $D \subseteq S_r$ are invariant under all rotations of the sphere surface, in the sense that for any rotation $T$, $T^{-1}(D)=D$?
Intuitively, the answer seems obvious: $S_r$ and $\emptyset$ are the only subsets that match this description, and they do so trivially. But I am not sure how to go about proving that they are the only ones. I am also worried that there may be some exotic subsets that also match the description.
(For what it's worth, I would also be interested in answers that restrict attention to subsets $D$ that belong to the Borel sigma algebra of $S_r$.)