I was reading these notes and on page 49, without concern the author writes about $X$, a random vector on the Euclidean sphere in $\mathbf{R}^n$, that is spherically distributed. He even writes: $$ X \sim \mathrm{Unif}(\sqrt{n}S^{n-1}). $$
The following question succeeds it:
Show that the spherically distributed random vector $X$ is isotropic. Argue that the coordinates of $X$ are not independent.
I realized I didn't understand what the distribution was exactly when I tried to solve the problem. There is a footnote in the text which makes it seem like $\mathbf{P}(X^{-1}(A)) = \lambda(A)/\lambda(S^{n-1})$, where $\lambda$ is the $n-1$-dimensional Lebesgue measure, and $A$ is a borel subset. I this true? Also, is there an explicit way to see the claim in the question? It would suffice to show that $\mathbf{E}\langle X, x\rangle^2 = \|x\|^2$ for every vector $x \in \mathbf{R}^n$.