TITLE: Volume of fundamental domain and Haar measure
QUESTION [14 upvotes]: In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. But I should confess I never felt confident with the Haar measure, essentially because theoretical part did not give me how to construct the Haar measure. Even when I was trying to understand Haar measure on $SL(n,\mathbb{R})$ via Iwasawa decomposition, I could not realize exactly how it works.
I think, or I should say I fell, I am not the only one how has problem with this delicate subject. So I thought I would want to share this with you and see how people think about the Haar measure and how you compute, for instance, the volume $${\rm Vol}(SL(n,\mathbb{Z})\backslash SL(n,\mathbb{R}))$$

REPLY [8 votes]: Supplementing the other answer (too long for a comment...) First, for examples of innocent-context computations, there is an on-line computation of volumes of $SL(n,\mathbb Z)\backslash SL(n,\mathbb R)$ and of $Sp(n,\mathbb Z)\backslash Sp(n,\mathbb R)$ 
 here , written in essentially Siegel's style. The same style of computation can be done adelically, over arbitrary number fields, but still does effectively beg the question of normalization. Nevertheless, such computations show that the normalization can be determined inductively, and, in any case, that the global computation can be done nicely once we have a locally-everywhere normalization of measures.
Yet-another approach is (after Langlands) to look at suitable residues of Eisenstein series (e.g., as in the Boulder conference, AMS Proc Symp IX). In effect, such a computation back-handedly normalizes the Haar measure...