Criterion for an invariant quotient measure

Assume $\nu$ is a nonzero invariant Radon measure on $X=G\mathrm{/}H$. Define $L(f)={\int }_X{T}_{H}fd\nu$ for real $f\in C_c(G)$. This functional is positive. If it were zero, surjectivity [F1] would make every $C_c(X)$ integral against $\nu$ zero, and [F2] would force $\nu =0$. Thus $L$ is nonzero.

Conversely suppose the modular functions agree on $H$. Then $\rho =1$ satisfies the covariance in [F5]. Let ${\mu }_1$ be the Weil measure. For $a\in G$ and $\varphi =T_{H}f$, its translate satisfies ${\int }_X\varphi (a^{-1}q)d{\mu }_1(q)={\int }_{G}f(a^{-1}x)dx={\int }_{G}f(x)dx={\int }_X\varphi (q)d{\mu }_1(q),$ by left invariance.

For $a\in G$, let $L_{a}f(x)=f(a^{-1}x)$. Then $T_H(L_{a}f)(xH)=T_{H}f(a^{-1}xH)$, so invariance of $\nu$ gives $L(L_{a}f)=L(f)$. By [F2], $L$ is represented by a Radon measure $\lambda$ on $G$; it is left invariant and nonzero, hence a left Haar measure.

By [F3], $\lambda =cdx$ for $c>0$. For $h\in H$, right translation gives $T_H(R_{h}f)={\mathrm{\Delta }}_H(h{)}^{-1}{T}_{H}f$ by [F4] applied in $H$. Hence $L(R_{h}f)={\mathrm{\Delta }}_H(h{)}^{-1}L(f)$. Since $\lambda =cdx$, [F4] applied in $G$ also gives $L(R_{h}f)={\mathrm{\Delta }}_G(h{)}^{-1}L(f)$. Choose $f$ with $L(f)>0$; equality forces ${\mathrm{\Delta }}_G(h)={\mathrm{\Delta }}_H(h)$.

Surjectivity [F1] gives this equality for every $C_c(X)$ test function. The Radon uniqueness in [F2] shows $a_{\ast }{\mu }_1={\mu }_1$; the Weil measure is nonzero. This proves sufficiency and the equivalence. ∎