Dense finitely generated subgroups and integration on compact groups (Q2515195)

Let \(G\) be a compact group, \(\mu\) the Haar measure on \(G\) and \(\rho : h \in G \mapsto \rho (h) \in \mathrm{End}_{\mathbb{C}} (V)\) a representation of \(G\) in the space \(V= C(G)= \{ f : g \in G \mapsto f(g) \in \mathbb{C} \;| \;f \;\mathrm{is} \;\mathrm{continuous} \;\}\) of all continuous complex functions on \(G\), realized by right shifts, that is, by \(\rho (h) \times f(g) = f (g * h)\). Theorem 1 of this short and elegant paper describes the integral \(\int_G f(g) \;d \mu\) as limit of the series with general term \(\mu_h f(h)\) (roughly speaking, \(\mu_h\) is a suitable restriction of \(\mu\) on a finitely generated dense subgroup, see p. 654 for the precise definition). An interesting formula of integration is described in Theorem 2, where the ratio is involved between the trace of an arbitrary element of \(\mathrm{End}_{\mathbb{C}} (V)\) and the degree of the representation.