This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very theoretical setting: in the right setting, it exists, it is unique (if the setting is really nice), and you can integrate against it to define new objects that will have nice properties because the measure itself does.

So my question is: how practical is it to compute with? I'm talking about very concrete examples here, e.g. "G=O(4), I integrate f(M)=[some explicit function with a matrix input] and the answer I get is 42". From conversations, I got the feeling that the construction of the Haar measure allows you in principle to write such a computation explicitly. I'm concerned with the tractability of the computation itself. Examples would be great.

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# Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very theoretical setting: in the right setting, it exists, it is unique, and you can integrate against it to define new objects that will have nice properties because the measure itself does.

So my question is: how practical is it to compute with? I'm talking about very concrete examples here, e.g. "G=O(4), I integrate f(M)=[some explicit function with a matrix input] and the answer I get is 42". From conversations, I got the feeling that the construction of the Haar measure allows you in principle to write such a computation explicitly. I'm concerned with the tractability of the computation itself. Examples would be great.