Construct in $\Bbb R^k$ a random variable $X$ that is uniformly distributed over the surface of the unit sphere in the sense that $|X|=1$ and $UX$ has the same distribution as $X$ for orthogonal transformations $U$. Hint Let $Z$ be uniformly distributed in the unit ball in $\Bbb R^k$, define $\phi(x) = \frac{x}{\|x\|}$ and take $X=\phi(Z)$.
I understand the intuition behind the solution i.e. orthogonal transform does not change the angles between points therefore the area of the $\phi$ preimage of the orthogonal transformed set on the surface remains unchanged. But, how do I formalize it ?
