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!