Corollary 3

Let $H$ be a locally compact topological group such that $G$ is a compact subgroup of $H$. Let $\varrho$ be a unitary representation of $H$ in a Hilbert space $E$. If the $\pi$-isotypical component of the restriction of $\varrho$ to $G$ is finite-dimensional for every $\pi \in \widehat{G}$, then the representation $\varrho$ of $H$ is semisimple and every irreducible unitary representation of $H$ occurs with finite multiplicity in $\varrho$.

Let us denote by $\sigma$ the restriction of $\varrho$ to the compact subgroup $G$ of $H$. Let $\pi$ be an irreducible representation of $G$. The endomorphism $\sigma (\chi_{\pi})\in \mathscr{L}(E)$ is of finite rank, since its image is the $\overline{\pi}$-isotypical component of $\sigma$ (cor. 2 of V, p. 465) and this is finite-dimensional by hypothesis. Consequently, the endomorphism $\sigma (f)$ is of finite rank for every $f\in \Theta (G^{\sharp})$ and is compact for every $f\in \mathscr{C}(G^{\sharp})$ (cor. 1 of V, p. 468 and cor. of prop. 2 of III, p. 4).

Let $j$ denote the canonical injection of $G$ into $H$. It is a continuous mapping which is $\nu$-proper for every measure $\nu$ on $G$ (INT, V, p. 69, § 6, No.$^o1$, Remark 1). Let $\mathfrak{B}$ be the filter of compact neighbourhoods of the element $e$ in $G$. For every $V\in \mathfrak{B}$, there exists a positive continuous central function $f_V$ on $G$ with support contained in $V$ whose integral over $G$ is equal to $1$. Let $\beta_V$ denote the image measure $j(f_V\cdot \mu)$; it is a positive measure with compact support on $H$ such that $\varrho (\beta_V) =\sigma (f_V)$ (INT, V, p. 71, § 6, No.$^o2$, Theorem 1). From the foregoing, the filter basis on $\mathscr{M}_c(H)$ which is the image of $\mathfrak{B}$ under the mapping $V\mapsto \beta_V$ satisfies the conditions of Proposition 2 of V, p. 402, and the assertion follows.

Corollary 4 (Weyl's criterion for equidistribution)

Let $M$ be a set of positive central measures on $G$ such that $\nu (G) = 1$ for every $\nu \in M$. Let $\mathfrak{F}$ be a filter on $M$. In order that the filter $\mathfrak{F}$ converge vaguely to the measure $\mu^{\sharp}$ on $G^{\sharp}$, it is necessary and sufficient that, for every non-trivial irreducible unitary representation $\pi$, one have

lim$_{\nu ,\mathfrak{F}}\int_G\chi_{\pi}(x)d\nu (x) = 0$.

Since $\nu (G^{\sharp}) = 1 =\mu^{\sharp}(G^{\sharp})$ for $\nu \in M$, the hypothesis means that

lim$_{\nu ,\mathfrak{F}}\int_{G^{\sharp}}\chi_{\pi}(x)d\nu (x) =\int_{G^{\sharp}}\chi_{\pi}(x)d\mu^{\sharp}(x)$

for every representation $\pi \in \widehat{G}$, hence it is equivalent to the condition

lim$_{\nu ,\mathfrak{F}}\int_{G^{\sharp}}f(x)d\nu (x) =\int_{G^{\sharp}}f(x)d\mu^{\sharp}(x)$

for every function $f\in \Theta (G^{\sharp})$ by linearity. Since the space $\Theta (G^{\sharp})$ is dense in $\mathscr{C}(G^{\sharp})$ (Corollary 1), the hypothesis is therefore equivalent to the convergence of the filter $\mathfrak{F}$ to $\mu^{\sharp}$ in $\mathscr{M}(G^{\sharp})$ endowed with the topology of simple convergence in $\mathscr{C}$ (G), which coincides with the topology of vague convergence since G is compact (cf. INT, III, p. 59, § 1, No.$^o9$).