Conjugation averaging on a compact group ($\mathcal{A}f(g)={\int }_{G}f(hg{h}^{-1})dh$)

Normalized Haar measure makes this averaging a norm-one projection onto continuous class functions. Averaging a trace coefficient replaces its coefficient endomorphism by $\int \theta (h{)}^{-1}\alpha \theta (h)dh$. On an irreducible summand this average is $(\mathrm{tr}\alpha /\dim V)I$ by Schur lemma. It follows that averaging finite-dimensional matrix coefficients yields finite linear combinations of irreducible characters. Together with the Peter-Weyl theorem, this proves uniform density of those characters among continuous class functions.