Existence of rho-functions and quotient measure classes

Define $\rho (x)={\int }_H\beta (xh){\mathrm{\Delta }}_G(h)\mathrm{/}{\mathrm{\Delta }}_H(h)dh$. For each $x$, the integrand is supported on the compact fiber intersection $x^{-1}\mathrm{supp}\beta \cap H$, so its integral is finite. It is positive because $\beta \ge 0$, the weight is positive, and ${\int }_H\beta (xh)dh=1$.

Near $x_0$ choose a compact neighborhood $K$. The set $S=\mathrm{supp}\beta \cap p^{-1}(p(K))$ is compact by [F2], and all $h$ for which $xh\in \mathrm{supp}\beta$ with $x\in K$ lie in the compact set $K^{-1}S\cap H$. The integrand is jointly continuous with this common compact support; [F4] gives continuity of its integral. Thus $\rho$ is positive and continuous.

For $h_0\in H$, substitute $k=h_{0}h$; left invariance of $dh$ and the homomorphism laws give $\rho (x{h}_0)={\mathrm{\Delta }}_H(h_0){\mathrm{\Delta }}_G(h_0{)}^{-1}\rho (x)$. Hence $\rho$ is a rho-function.

Apply [F5] to obtain ${\mu }_{\rho }$ and the Weil formula. The ratio $D_g(xH)=\rho (g^{-1}x)\mathrm{/}\rho (x)$ is independent of the representative by [F3] and is positive continuous. For $\varphi =T_{H}f$, Weil and left invariance give ${\int }_{G\mathrm{/}H}\varphi (gq)d{\mu }_{\rho }(q)={\int }_{G}f(gx)\rho (x)dx={\int }_{G}f(y)\rho (g^{-1}y)dy={\int }_{G\mathrm{/}H}\varphi (q)D_g(q)d{\mu }_{\rho }(q)\mathrm{.}$ By [F6] this holds for every $\varphi \in C_c(G\mathrm{/}H)$, and [F7] identifies $g_{\ast }{\mu }_{\rho }=D_g{\mu }_{\rho }$. Positivity of $D_g$ gives equivalence of measures; the ratio descends continuously jointly in $(g,q)$ through the open quotient map. Thus ${\mu }_{\rho }$ is strongly quasi-invariant. Full support is part of [F5]. ∎