I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz).
Right from the very beginning, Weil uses the fact that when you have a locally compact topolgocal group $G$ and a locally compact subgroup $H$, in addition to the Haar measures on $G$ and $H$, there exists a "Haar measure" on the coset space $G/H$, with some properties.
For instance, the upper half plane $\mathbb H$ is the quotient $SL_2(\mathbb R)/SO_2(\mathbb R)$ and the usual measure there which gives rise to the usual hyperbolic metric, is arising in this way.
I originally assumed this theorem and went ahead(but not much) with that book.
I want to have a reference for the above theorem. A reference which is not written by Weil. I find him very hard to penetrate. This should exclude Bourbaki's "Integration", as I supppose it would be heavily influenced by him, and thus is a horrible book(note to Harry: this is personal opinion; spare me the brickbats).
I had originally seen the construction of Haar measure on H. Royden's "Real Analysis", in which he is not considering any quotients.
Best Answer
The book I always look at for such things is Nachbin, The Haar Integral, which is short, and has a whole chapter on Integration on Locally Compact Homogeneous Spaces.
And a plus: he gives you a choice of reading the proof of the existence and uniqueness of the Haar integral according to Weil or according to Henri Cartan.