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. https://mathoverflow.net/questions/4939/is-there-a-compact-group-of-countably-infinite-cardinality/4950#4950); but I cannot really say the same thing about the first one.

I would love to expand this list!