Example

— 1) One can take $f \in \mathscr{C}(X)$, $\nu \in \mathscr{C}'(X)$ and $\mu \in \mathscr{C}'(G)$ by Prop. 3, and the Cor. of Prop. 5 of §1, No. 4. The formula (7) then means that the endomorphism $\nu \mapsto (\chi \cdot \check{\mu}) * \nu$ of $\mathscr{C}'(X)$ is the transpose of the endomorphism $f \mapsto \mu * f$ of $\mathscr{C}(X)$.

2) One can take $f \in \mathscr{K}(X)$, $\nu \in \mathscr{M}(X)$ and $\mu \in \mathscr{C}'(G)$ by Prop. 3, Prop. 8 of §3, No. 2, and the remark that the support of the continuous function $g(s, x)$ intersects the support of $\mu \otimes \nu$ in a compact set. The formula (7) then means that the endomorphism $\nu \mapsto (\chi \cdot \check{\mu}) * \nu$ of $\mathscr{M}(X)$ is the transpose of the endomorphism $f \mapsto \mu * f$ of $\mathscr{K}(X)$.

3) If $G$ operates properly on $X$, one can take $f \in \mathscr{K}(X)$, $\nu \in \mathscr{C}'(X)$ and $\mu \in \mathscr{M}(G)$ by Prop. 4, Prop. 8 of §3, No. 2, and the same remark as in Example 2.