A homogeneous quotient without invariant measure

The subgroup $H=\{(0,a):a>0\}$ is closed because it is the zero set of the continuous first-coordinate map, and it is a subgroup by the multiplication law. The map $(b,a)\mapsto b$ is continuous and constant on right $H$-cosets; its fibers are exactly those cosets, since $(b,a)(0,t)=(b,at)$. It therefore descends to a continuous bijection $\bar{q}:G\mathrm{/}H\to \mathbb{R}$. The inverse $b\mapsto (b,1)H$ is continuous as a composition of the continuous section $b\mapsto (b,1)$ with the quotient map, so $\bar{q}$ is a homeomorphism. Under this identification the left action of $(b_0,a_0)$ is $T(x)=b_0+a_{0}x$. Let $m$ be a left Haar Radon measure on $(\mathbb{R},+)$, supplied by [F1]. For each such $T$, the measure ${\nu }_T(E)=m(T^{-1}E)$ on Borel sets is Radon because $T$ is a homeomorphism. It is translation invariant: $T^{-1}(E+u)=T^{-1}(E)+u\mathrm{/}{a}_0$, so left invariance of $m$ applies. Thus [F2] gives ${\nu }_T=c_{T}m$ for some $c_T>0$. Hence every group translate scales $m$ by a positive finite constant, so its null sets are preserved and its Radon measure class is quasi-invariant.

On the two-dimensional group manifold use the measure $d{\mu }_G(b,a)=a^{-2}dbda$, defined by the positive continuous density in [F3] relative to two-dimensional Lebesgue measure. By [A1, A2], AC supplies countable choice, so [F8] makes the base measure Radon; the density is bounded above on each compact set, so the weighted measure is locally finite and inherits regularity from the base measure. Left multiplication by $(b_0,a_0)$ sends $(b,a)$ to $(b_0+a_{0}b,a_{0}a)$ with Jacobian $a_0^2$; substituting these coordinates gives $a^{-2}dbda=(a_{0}a{)}^{-2}d(b_0+a_{0}b)d(a_{0}a)$, so ${\mu }_G$ is left invariant. Therefore it is a left Haar measure by [F7]. For $h=(0,t)\in H$, right multiplication sends $(b,a)$ to $(b,at)$; with $A=at$, $a^{-2}dbda=t{A}^{-2}dbdA$, and hence ${\int }_{G}f(xh)d{\mu }_G(x)=t{\int }_{G}f(x)d{\mu }_G(x)$ for $f\in C_c(G)$. Applying [F4] to $h^{-1}=(0,t^{-1})$ yields ${\mathrm{\Delta }}_G(0,t)=t^{-1}$. On the other hand [F5] gives ${\mathrm{\Delta }}_H(0,t)=1$, since $H$ is abelian. Taking any $t\mathrm{\ne }1$, the modular functions disagree on $H$; [F6] rules out every nonzero invariant Radon measure on $G\mathrm{/}H$. ∎