Defining a Measure on Quotient Spaces Let $G$ be a locally compact Hausdorff group with a left invariant Haar measure $\mu$ and a closed subgroup $H$. It is well-known (and not hard to prove) that $G/H$ possesses an invariant measure if and only if the modular functions of $G$ and $H$ agree on $H$.
I am looking for the exact reason for which the following naive approach to defining such a measure on $G/H$ fails.
Let $f\in C_c(G/H)$ and define
$$\int_{G/H} f\, d\sigma:=\int_G \chi f\circ\pi \, d\mu$$
where $\pi\colon G\to G/H$ is the canonical projection map and $0\le \chi\in C_c(G)$ has the property that $\int_G \chi d\mu=1$ (or perhaps with some similar property.)
Assuming that $G/H$ does have an invariant measure, can it be expressed in the above form?
 A: I think the impulse to "lift" a function on $G/H$ to $G$, and define the integral on $G/H$ as the integral on $G$ of that lift, is entirely reasonable... but doesn't quite work out, no matter how one struggles. A prescription for seeing the problem in various (futile) attempts resides in the key point for the successful characterization: letting $\alpha$ be the averaging map $(\alpha f)(x)=\int_H f(hx)\,dh$, with the sided-ness of the measure easily determined... the requirement is exactly as suggested in the question:
$$
\int_G f(g)\,dg \;=\; \int_{G/H} (\alpha f)(\dot{g})\,d\dot{g}
$$ 
where the $\dot{g}$ is the coordinate on the quotient. To make this be an unequivocal characterization of the integral/measure on the quotient, one must check that the averaging map is a surjection of $C^o_c(G)$ to $C^o_c(G/H)$, which is indeed so, in general.
Then, to return to the question of making a "section" of the averaging map... if the condition on "modular functions" is met, and if continuity is not necessary, then this is certainly possible. But this might be a less than optimal approach, since we might want to characterize measures/integrals via Riesz' theorem about functionals on $C^o_c$.
NB, as @KConrad suggests, the modular-function condition is necessary, as one fills out the alleged proof that such a formula gives an invariant measure. The popular cases, such as $\mathbb R \to \mathbb R/\mathbb Z$ with section given by multiplying by characteristic function of an interval, so subliminally assume the modular-function condition that we may not have realized.
A: For general quotient, you will have to work with quasi-invariant measures. There is a multiplier, which is constant if the measure is invariant. For it to be constant, it is necessary and sufficient that $\Delta_G|_{H} = \Delta_H$.
