Proposition 4

The measure $dz$ on $\mathfrak{g}$ is the image under the proper map $(g, x) \rightarrow$ (Ad $g$)$(x)$ from $G\times \mathfrak{t}$ to $\mathfrak{g}$ of the measure $dg\otimes \frac{1}{w(G)}\pi_{\mathfrak{g}}dx$.

We leave to the reader the statement and proof of the analogues of Cor. 1 to 3 and Remarks 1 to 3 of no. 2. For example, let $\varphi$ be an integrable function on $\mathfrak{g}$ (with values in a Banach space or $\overline{\mathbf{R}}$); then

$\int_{\mathfrak{g}}\varphi (z)dz=\frac{1}{w(G)}\int_{\mathfrak{t}}(\int_G\varphi$((Ad $g$)$x$)$dg)\pi_{\mathfrak{g}}(x)dx$, (12)

and, in particular, if $\varphi$ is invariant under Ad(G),