The following proposition appears in page 141 in Knapp's book, representation theory of semisimple groups.
Let $G$ be linear connected reductive, and fix a positive system $\Sigma^+$ of restricted roots. For suitable normalizations of Haar measures, with $\mathfrak{a}^+$ as the positive Weyl chamber, and for $f\in C_{\mathrm{com}}(G)$, $$ \int_G f(x) dx=\int_{K\times\mathfrak{a}^+\times K}\left[\prod_{\lambda\in\Sigma^+}(\sinh\lambda(H))^{\mathrm{dim}\mathfrak{g}_\lambda}\right] f(k_1\exp(H)k_2)\mathrm{d}k_1\mathrm{d}H\mathrm{d}k_2 $$
Where can I find a proof of the above formula?
They claim to give a proof (c.f. theorem 2.6): http://arxiv.org/abs/math/0504220