Suppose $S$ is the sphere of radius one. And suppose $f:S\to \mathbb{R}$ is defined as follows: $$f(x_1,...,x_n)=\frac{1}{x_1^2}+\frac{1}{x_2^2}+\cdots+\frac{1}{x_n^2}$$
I am trying to calculate either of these: $$\int f(\gamma)d\gamma$$ where integrating is with respect to Haar measure over the sphere. Or maybe finding the median of the induced random variable on $\mathbb{R}$ by $f$. I appreciate any help in this direction.
Let's use spherical coordinates:
$$ x_k = \prod_{j=1}^{k-1} \sin(\phi_j) \cos(\phi_k) \\ J = \prod_{j=1}^{n-2} \sin^{n-1-j}(\phi_j) $$
I omitted $r$ because we'll be considering sphere of unit radius. Thus
$$ \int f d S = 2^n \int_{0}^{\pi/2} \dots \int_{0}^{\pi/2} \Bigl( \sum_{k=1}^n \prod_{j=1}^{k-1} \sin^{-2}(\phi_j) \cos^{-2}(\phi_k) \Bigr) \prod_{j=1}^{n-2} \sin^{n-1-j}(\phi_j) d \phi_1 \dots d \phi_n \\ = 2^n \sum_{k=1}^n \int_{0}^{\pi/2} \dots \int_{0}^{\pi/2} \prod_{j=1}^{k-1} \sin^{-2}(\phi_j) \cos^{-2}(\phi_k) \prod_{j=1}^{n-2} \sin^{n-1-j}(\phi_j) d \phi_1 \dots d \phi_n $$
The last equation looks frightening, but it's just a product on $n$ integrals of sinuses and cosinuses raised to some power. And some of them are of particular interest.
Consider $\phi_{n-1}$ for $k = n$. Corresponding integral is
$$ \int_0^{\pi/2} \sin^{-2}(\phi_{n-1}) d\phi_{n-1} $$
This integral is divergent. Clearly, the whole sum thus is unbounded.
Thus the expectation does not exist.
