Dense finitely generated subgroups and integration on compact groups

We point out methods of approximation of the commutator Hamiltonian on a compact group $G$ with finite sums of the form $\sum\limits_{g\in G}\sum\limits_{h\in G}\mu_g\nu_hghg^{-1}h^{-1}$, where $\sum\limits_{g\in G}\mu_g=1$ and $\sum\limits_{h\in G}\nu_h=1$.