Density of averaged covariant generators

For fixed $x$, the integrand defining ${\xi }_{f,v}$ is supported on the compact set $x^{-1}\mathrm{supp}f\cap H$, so the Bochner integral exists. Replacing $x$ by $x{h}_0$ and substituting $k=h_{0}h$ gives ${\xi }_{f,v}(x{h}_0)=\sigma (h_0{)}^{-1}{\xi }_{f,v}(x)$. For a relatively compact neighborhood $N$ of a fixed $x_0$, every contributing $h$ lies in the compact set $K={\bar{N}}^{-1}\mathrm{supp}f\cap H$. The integrand $(x,h)\mapsto f(xh)\sigma (h)v$ is jointly continuous on a compact neighborhood times $K$, so its uniform variation in $h$ tends to zero as $x\to x_0$; the integral therefore varies continuously. Its quotient support lies in $p(\mathrm{supp}f)$, which is compact. Thus ${\xi }_{f,v}\in C_c(G,H;V)$.

Now let $F$ be a locally strongly measurable covariant section with finite quotient $L^2$ norm. By [F5] choose $\psi \in C_c(G\mathrm{/}H)$ close in scalar $L^2$ to $q\mapsto \mathrm{\parallel }F(q)\mathrm{\parallel }$; outside $Q=\mathrm{supp}\psi$ the $L^2$ tail of $F$ is therefore small. Put $F_Q={\mathbf{1}}_{Q}F$ and choose a cutoff $\chi \in C_c(G\mathrm{/}H)$ with $\chi =1$ on $Q$ by [F3], then choose a nonnegative lift $u_0\in C_c(G)$ with $T_H{u}_0=\chi$ by [F3]. The measurable map $U=u_0{F}_Q$ is supported in a compact subset of $G$. For $\varphi \in C_c(G\mathrm{/}H)$, apply the Weil formula [F8] to $f(y)=(\varphi \circ p)(y)\mathrm{\mid }{u}_0(y){\mathrm{\mid }}^2$. It identifies the finite Radon measures $B\mapsto {\int }_{p^{-1}(B)}\mathrm{\mid }{u}_0(y){\mathrm{\mid }}^2\rho (y)dy$ and $B\mapsto {\int }_B{T}_H(\mathrm{\mid }{u}_0{\mathrm{\mid }}^2)(q)d{\mu }_{\rho }(q)$, first on $C_c(G\mathrm{/}H)$ and then on Borel sets by [F8]. Integrating $q\mapsto \mathrm{\parallel }{F}_Q(q){\mathrm{\parallel }}^2$ gives ${\int }_G\mathrm{\parallel }U(x){\mathrm{\parallel }}^2\rho (x)dx={\int }_Q\mathrm{\parallel }F(q){\mathrm{\parallel }}^2{T}_H(\mathrm{\mid }{u}_0{\mathrm{\mid }}^2)(q)d{\mu }_{\rho }(q)<\mathrm{\infty }\mathrm{.}$ On its compact support $\rho dx$ is finite, so [F6] makes $U$ Bochner square integrable.

Let $F\in C_c(G,H;V)$ and $K={\mathrm{supp}}_{G\mathrm{/}H}F$. Choose a cutoff $\chi \in C_c(G\mathrm{/}H)$ with $\chi =1$ on $K$ by [F3], then a nonnegative lift $u_0\in C_c(G)$ with $T_H{u}_0=\chi$ by [F3]. The map $u_{0}F$ is continuous and compactly supported on $G$. Cover its compact support by finitely many open sets on which $F$ varies by less than $ϵ$ in norm; [F4] supplies a subordinate partition ${\theta }_j$. For chosen $v_j$ from each patch, $u_{0}F$ is uniformly within $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}ϵ$ of ${\sum }_j(u_0{\theta }_j)v_j$. Averaging the latter gives a finite sum of generators. The averaging error is bounded uniformly on the compact quotient support because, after choosing a compact lift $C$ of that support, all relevant $h$ lie in the fixed compact set $C^{-1}\mathrm{supp}{u}_0\cap H$, of finite Haar measure. Since $A(u_{0}F)=T_H{u}_{0}F=F$ on $K$ and both vanish off $K$, the generators approximate $F$ uniformly.

Strong measurability approximates $U$ by finite-valued simple maps; scalar $C_c$ density [F5] approximates their coefficients in $L^2(G,\rho dx)$. Multiplying by one fixed compactly supported cutoff equal to one on $\mathrm{supp}U$ makes all approximants supported in a common compact $C$. The averaging operator $A(W)(x)={\int }_H\sigma (h)W(xh)dh$ is bounded on continuous maps supported in $C$. If $C=\mathrm{\varnothing }$ then $W=0$ and the bound is immediate. Otherwise choose a compact lift $C_0$ of $p(C)$ and let $m=dh(C_0^{-1}C\cap H)<\mathrm{\infty }$. For each $q\in p(C)$ choose a representative $x\in C_0$. Cauchy–Schwarz gives $\mathrm{\parallel }A(W)(x){\mathrm{\parallel }}^2\le m{T}_H(\mathrm{\parallel }W{\mathrm{\parallel }}^2)(q)$. Integrating over $G\mathrm{/}H$ and applying [F8] to $\mathrm{\parallel }W{\mathrm{\parallel }}^2\in C_c(G)$ yields $\mathrm{\parallel }A(W){\mathrm{\parallel }}_2^2\le m{\int }_G\mathrm{\parallel }W(y){\mathrm{\parallel }}^2\rho (y)dy=m\mathrm{\parallel }W{\mathrm{\parallel }}_{L^2(G,\rho dy)}^2\mathrm{.}$ Therefore averages of the finite-sum $C_c(G,V)$ approximants converge to $A(U)=F_Q$. Each average is a finite sum of the generators in step 1.1. Letting the discarded tail tend to zero proves density in the full measurable $L^2$ space. ∎