TITLE: Haar Measure on a Quotient
QUESTION [8 upvotes]: Suppose you have a locally compact group G with a discrete subgroup H.  Of course G has a unique (up to scalar) Haar measure, but it seems that G/H has and induced Haar measure as well.  
How does one induce a measure on G/H from the Haar measure on G? Any sources that describe this?

REPLY [7 votes]: I couldn't figure out which other answer to make this be a comment for... so: 
Already mentioned in other answers, but really working better than might have been acknowledged, is that the averaging map $\alpha:C^o_c(G)\rightarrow C^o_c(H\backslash G)$ by $\alpha f(x)=\int_H f(hx) dh$ is (readily provably) a surjection, for a closed subgroup $H$ of $G$. Thus, to attempt to define a right $G$-invariant integral/measure on $C^o_c(H\backslash G)$, define $\int_{H\backslash G} \alpha f dx=\int_G f$. The immediate issue, of course, is well-definedness, which holds only when the modular function of $G$ restricted to $H$ is equal to the modular function of $H$. 
This is one of those quasi-standard riffs that gets submerged... perhaps I saw it in Weil's book on integration on topological groups.