Proposition 4

Assume that G operates properly in X and that $\chi$ is continuous. Let $\mu \in \mathscr{M}(G)$ and $f \in \mathscr{K}(X)$.

(i) $\mu$ and $f$ are convolvable relative to $\beta$.

(ii) Formula (3) of No. 1 defines for every $x \in X$ a convolution product $\mu *^\beta f$ that is continuous.

(iii) The mapping $(\mu, f) \mapsto \mu *^\beta f$ of $\mathscr{M}(G) \times \mathscr{K}(X)$ into $\mathscr{C}(X)$ is hypocontinuous relative to the bounded subsets of $\mathscr{M}(G)$ and the compact subsets of $\mathscr{K}(X)$ that are contained in some subspace $\mathscr{K}(X, L)$ (where $L$ is a variable compact subset of $X$).

We know that $\mu$ and $f$ are convolvable (§ 3, No. 2, Prop. 8 (ii)), and it is clear that the integrals occurring in (3) exist for every $x \in X$. Let $K$ and $L$ be two compact subsets of $X$. There exists a compact subset $H$ of $G$ such that the relations $x \in K$ and $s^{-1}x \in L$ imply $s \in H$; let $\varphi \in \mathscr{K}_+(G)$ with $\varphi(s) = 1$ for $s \in H$. Then, for $f \in \mathscr{K}(X, L)$ and $x \in K$,

Consequently $\int f(s^{-1}x)\chi(s^{-1}, x)\, d\mu(s)$ is a continuous function of $x$ and defines a convolution product $\mu *^\beta f \in \mathscr{C}(X)$. Moreover, the mapping $\mu \mapsto \varphi \cdot \mu$ of $\mathscr{M}(G)$ into $\mathscr{C}'(G)$ is continuous for the topologies of compact convergence. Prop. 3 (iii) therefore implies that the mapping $(\mu, f) \mapsto \mu *^\beta f$ of $\mathscr{M}(G) \times \mathscr{K}(X, L)$ into $\mathscr{C}(X)$ is, for every compact subset $L$ of $X$, hypocontinuous relative to the compact subsets of $\mathscr{K}(X, L)$. In particular, the mapping $(\mu, f) \mapsto \mu *^\beta f$ of $\mathscr{M}(G) \times \mathscr{K}(X)$ into $\mathscr{C}(X)$ is separately continuous. Since $\mathscr{K}(X)$ is barreled, this mapping is hypocontinuous relative to the bounded subsets of $\mathscr{M}(G)$ (TVS, III, § 5, No. 3, Prop. 6).