Theorem 1 (H. Weyl)

The measure $dg$ on G is the image under the map $(g, t) \rightarrow gtg^{-1}$ from $G\times T$ to G of the measure $dg\otimes \frac{1}{w(G)}\delta_Gdt$, where

$\delta_G(t) =$ det(Ad$_{\mathfrak{g}/\mathfrak{t}}(t)-1$) $=\prod_{\alpha\in R(G,T)}(t^{\alpha}-1)$.

Equivalently (Integration, Chap. V, §6, no. 3, Prop. $4$)$,dg$ is the image under the map $f: (G/T)\times T\rightarrow G$ of the measure $\mu\otimes \frac{1}{w(G)}\delta_Gdt$.

We prove the last assertion. It follows from §5, no. 1 and Differentiable and Analytic Manifolds, Results$, 10.1.3c)$ that G-- $G_r$ is negligible in G and T -- $T_r$ is negligible in T. Further, the map $f_r$ makes $(G/T)\times T_r$ a principal covering of $G_r$, with group W (§5, no. 4, Prop. $4b$)$)$. The theorem now follows from Lemma 3 and Integration, Chap. V, §6, no. 6, Prop. 11.