Exercise 8, § 3

a) Show that there exists on $S_{n-1}$ a measure $\omega_{n-1}$ invariant under $O(n, \mathbf{R})$ and, up to a constant factor, only one (regard $S_{n-1}$ as a homogeneous space for $O(n, \mathbf{R})$).

b) The mapping $(t, z) \mapsto tz$ permits identifying the topological space $\mathbf{R}^n - \{0\}$ with the topological space $\mathbf{R}_+^* \times S_{n-1}$. Show that the measure induced on $\mathbf{R}^n - \{0\}$ by the Lebesgue measure on $\mathbf{R}^n$ may then be identified, up to a constant factor, with $t^{n-1} dt \otimes d\omega_{n-1}(z)$. (Make use of the fact that the Lebesgue measure on $\mathbf{R}^n$ is invariant under $O(n, \mathbf{R})$ and the fact that the Lebesgue measure of the closed ball with center 0 and radius $r$ is proportional to $r^n$.)

c) Let $L_h$ be the intersection of $S_{n-1}$ and the hyperplane $x_n = h$. The non-empty $L_h$ are the intransitivity classes of $O(n-1, \mathbf{R})$ in $S_{n-1}$. Show that $\omega_{n-1}$ may be put in the form $\int \lambda_h d\nu(h)$, where $\lambda_h$ is a measure invariant under $O(n-1, \mathbf{R})$ and carried by $L_h$, and $\nu$ is a measure on $[-1, 1]$. Identifying $L_h$ with $S_{n-2}$ by $z \mapsto he_n + \sqrt{1-h^2} z$, one can assume that $\lambda_h = \omega_{n-2}$. Let $K_\theta$ be the subset of $S_{n-1}$ defined by $\sin \theta \leq x_n \leq 1$. By calculating the Lebesgue measure of the set of $tz$ ($0 \leq t \leq 1$, $z \in K_\theta$), show that $\omega_{n-1}(K_\theta)$ is proportional to $\int_\theta^{\pi/2} \cos^{n-2} \varphi \, d\varphi$. From this, deduce that if one sets $h = \sin \theta$ ($-\pi/2 \leq \theta \leq \pi/2$), then the measure $\nu$ may be identified with $\cos^{n-2} \theta \, d\theta$.

d) Conclude, by induction on $n$, that

where

with $-\pi/2 \leq \theta_i \leq \pi/2$ for $1 \leq i \leq n-2$, $0 \leq \theta_{n-1} < 2\pi$.