Let $G$ be a compact connected Lie group, let $T\le G$ be a maximal torus, and let $W:=N_G(T)/T$ be the finite Weyl group, acting on $T$ by conjugation. Let $\mu_G$ and $\mu_T$ denote the normalized Haar probability measures on $G$ and $T$, respectively. Let $X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,S^1)$, let $\Phi\subset X^*(T)$ be the root system of the complexified [Lie algebra](/page/Lie%20Algebra) of $G$ with respect to $T$, choose a positive root system $\Phi^+\subset \Phi$, and define the Weyl denominator density $D:T\to [0,\infty)$ by

\begin{align*} D(t):=\prod_{\alpha\in\Phi^+}|1-\alpha(t)|^2. \end{align*}

Here each root $\alpha$ is regarded as a continuous character $\alpha:T\to S^1$.

Let $C\subset T$ be a measurable fundamental domain for the $W$-action on $T$ up to $\mu_T$-null sets, meaning that there is a measurable set $E\subset T$ with $\mu_T(E)=0$ such that

\begin{align*} T\setminus E=\bigsqcup_{w\in W}w(C\setminus E). \end{align*}

Then for every $\mu_G$-integrable measurable class function $f:G\to\mathbb C$, meaning $f(hgh^{-1})=f(g)$ for all $g,h\in G$, one has

\begin{align*} \int_G f(g)\,d\mu_G(g)=\int_C f(t)D(t)\,d\mu_T(t). \end{align*}