Let $G$ be a compact connected Lie group, let $T\le G$ be a maximal torus, and let

\begin{align*} W:=N_G(T)/T \end{align*}

be its finite Weyl group. Let $X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,S^1)$ be the character group of $T$. Let $R\subset X^*(T)$ be the finite root system of the complexified [Lie algebra](/page/Lie%20Algebra) of $G$ with respect to $T$, and choose a set $R^+\subset R$ of positive roots. Regard each root $\alpha\in R$ as a continuous character $\alpha:T\to S^1\subset\mathbb C$, and define the Borel function $D:T\to [0,\infty)$ by

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

Let $\mu_G$ and $\mu_T$ be the normalized Haar probability measures on the Borel spaces $G$ and $T$, respectively. If $f:G\to\mathbb C$ is $\mu_G$-integrable and admits a conjugation-invariant Borel representative $\tilde f:G\to\mathbb C$, then $\tilde f|_T\,D$ is $\mu_T$-integrable and, for every such representative $\tilde f$,

\begin{align*} \int_G \tilde f(g)\,d\mu_G(g)=\frac{1}{|W|}\int_T \tilde f(t)D(t)\,d\mu_T(t). \end{align*}