I am looking for examples of (not necessarily deep) results whose proofs rely on the Haar measure (or rather such that there exists an "elegant" proof involving it). Further, the formulation of such results should not refer to the Haar measure in any way, even implicitly.

I have the following couple of examples in mind :

  • The version of the Peter-Weyl theorem stating that one can decompose any unitary representation of a compact group as a direct sum of finite dimensional representations;

  • A compact (Hausdorff) group cannot be infinite countable.

I know there are many other elegant proofs of the second result out there not relying on the Haar measure at all (cf. Is there a compact group of countably infinite cardinality?); but I cannot really say the same thing about the first one.

I would love to expand this list!

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    $\begingroup$ The fact that every locally compact group has a faithful (continuous) unitary representation relies on the existence of the Haar measure, namely the left regular representation works. In turn, this is used (along with additional arguments) to show that (continuous) irreducible representations separate points. $\endgroup$ – YCor Mar 8 at 16:37
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    $\begingroup$ Does math.stackexchange.com/questions/61389/… work? $\endgroup$ – Steven Gubkin Mar 8 at 16:45
  • $\begingroup$ @StevenGubkin seems to qualify, thank you! $\endgroup$ – J.F Mar 8 at 16:53
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    $\begingroup$ Compact linear groups are the set of real points of an algebraic group, may qualify ? $\endgroup$ – user120527 Mar 12 at 16:40

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