Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 19G / i

Normalize Haar measure on $\mathrm{SU}(2)$ to total mass one. For a continuous class function $F$, the Weyl integration formula for SU2 isEquivalently it is ${\pi }^{-1}{\int }_0^{2\pi }F(t_{\theta }){\sin }^2\theta d\theta$. For arbitrary continuous $F$, replace $F(t_{\theta })$ on the right by its conjugacy average ${\int }_{\mathrm{SU}(2)}F(h{t}_{\theta }{h}^{-1})dh$.

Here is a direct proof. Identify $\mathrm{SU}(2)$ with unit quaternions, the sphere $S^3$. Multiplication by a unit quaternion is a transformation given by an orthogonal matrix of ${\mathbb{R}}^4$, so normalized round volume is bi-invariant and is Haar measure. Write a quaternion as $\cos \theta +\mathbf{n}\sin \theta$, $0\le \theta \le \pi$, $\mathbf{n}\in S^2$. The round volume element is ${\sin }^2\theta d\theta dA(\mathbf{n})$, and the total volume is $2{\pi }^2$. Conjugation rotates the unit imaginary vector $\mathbf{n}$, so a class function depends only on $\theta$. Integrating over the sphere of area $4\pi$ gives the factor $4\pi /(2{\pi }^2)=2/\pi$. The endpoint degeneracies have measure zero. Conjugacy averaging and Haar invariance prove the more general version.