Lemma 3

The inverse image on $(G/T)\times T_r$ of the measure $dg$ on $G_r$ under the local homeomorphism $f_r($Integration, Chap. V, §6, no. 6) is the measure $\mu\otimes \delta_Gdt$, where $\mu$ is the unique G-invariant measure on $G/T$ of total mass 1.

Choose an invariant differential form $\omega_T$ (resp. $\omega_{G/T}$) on T (resp. $G/T$) of maximal degree, such that the measure defined by $\omega_T$ (resp. $\omega_{G/T}$) is equal to $dt$ (resp. $\mu$). Put $\omega_G=\omega_{G/T}\cap \omega_T$. Lemma 2 implies that the measure defined by $\omega_G$ is equal to $dg$. Let U be an open subset of $(G/T)\times T_r$ such that $f_r$ induces an isomorphism from U to an open subset V of $G_r$. Let $\varphi$ be a continuous function with compact support in V; denote also by $\varphi$ the extension of $\varphi$ to $G_r$ which vanishes outside V. We have

$=\int_U(\varphi \circ f_r)\omega_{G/T}\wedge \delta_G\omega_T$ (Prop$.1$)

hence the lemma.