I've only really worked with real-valued random variables. But suppose $X$ is a random variable taking values in $\mathbb{S}^{n-1}$. What does it mean for $X$ to be uniformly distributed?
The distribution is just $P(X \in A)$ for $A \subset \mathbb{S}^{n-1}$. What does it mean for this to be a uniform distribution? I have a hunch that this means it behaves like the spherical measure $\sigma$ on $\mathbb{S}^{n-1}$. But I do not know the formal definition so I'd appreciate someone explaining that to me.
A related question is whether there is only one such uniform measure on the sphere, up to a constant multiple. It feels rather like the question of Haar measure on a locally compact Abelian group, but that's only for translation invariance. Here we're really talking about rotation invariance. That, after all, is what $\sigma$ is all about. Is there such a uniqueness statement for rotationally invariant measures on the sphere, and could one prove that using the uniqueness of Haar measure?
