Proposition 2

Let $\varrho$ be a unitary representation of G in a complex Hilbert space E. Let $\mathfrak{F}$ be a basis of filter on $\mathscr{M}_c(G)$ converging to the measure $\varepsilon_e$ for the topology of compact convergence. Suppose that there exists $M\in \mathfrak{F}$ such that $\varrho (\nu )$ is a compact endomorphism of E for every $\nu \in M$.

Then the unitary representation $\varrho$ is semisimple and every irreducible unitary representation of G has finite multiplicity in $\varrho$.

Let us first prove a lemma.