I'm trying to compute the Haar measures of finite unitary group $U(d)$ and finite orthogonal group $O(d)$. I've shown that both are compact groups. Most of computings I've seen base on Peter-Weyl Theorem but I'm not familiar with Lie Groups.

Is not possible to compute it by the exercise of Cohn's Measure Theory that takes place below :

If for each $a$ in $G$ the function $u \mapsto \varphi(a\varphi^{−1}(u))$ is the restriction to $U$ of an affine map $L_a : \Bbb R^d \to \Bbb R^d$, then the formula $\mu(A)=\int_{\varphi(A)}|\det(L_{\varphi^{−1}(u)}|\lambda(du)$ defines a left Haar measure on G. ($\lambda$ is Lebesgue measure on $\Bbb R^d$)

Can someone please explain how can I use this exercise for computing? Can I use Cayley transform as homeomorphism and do I have to use parametrization?

Thanks in advance


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