compact groups are unimodular
Theorem - If is a compact Hausdorff topological group, then is unimodular, i.e. it’s left and right Haar measures coincide.
Proof:
Let denote the modular function of . It is enough to prove that is constant and equal to , since this proves that every left Haar measure is right invariant.
Since is continuous and is compact, is a compact subset of . In particular, is a bounded subset of .
But if is not identically one, then there is a such that (recall that is an homomorphism). Hence, as increases, which is a contradiction
since is bounded.