I am reading Brian Hall's Lie Groups, Lie Algebras, and Representation Theory. His Prop. 4.36 reads
If $G$ is a compact matrix Lie group, $G$ has the complete reducibility property.
The "complete reducibility property" means that every finite representation (of $G$) is completely reducible (i.e., is equivalent to a direct sum of irreducible representations). His proof invokes Haar measure, which he (admittedly) does not prove the existence of. He also uses (again without proof) that the Haar measure of a compact group is finite.
Is there a proof that does not require Haar measure or integration on manifolds?
In particular, the fact that we are only interested in compact matrix Lie groups should simplify things. I found some sources that suggest that existence of Haar measure is "trivial" to establish on Lie groups, but this requires familiarity with volume forms on manifolds, and I would prefer to even avoid this.