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The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is semi-simple. Do you know any other nice applications of the Haar measure?

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In Tate's thesis, Hecke L-functions are written as an integral over the group of ideles. He then uses Fourier analysis over the adeles (itself an apllication of the Haar measure) to prove the functional equation for such L-functions. –  Jeff Tolliver Dec 7 '12 at 3:58
    

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One important "application" is Peter-Weyl theorem. A good article can be found at here.

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