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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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    http://mathoverflow.net/questions/4939/is-there-a-compact-group-of-countably-infinite-cardinality/495$0$#49502012-12-07

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