Remark

— 1) Let $\lambda_0 \in \mathscr{C}'(G)$. The mapping $\mu \mapsto \lambda_0 * \mu$ of $\mathscr{M}(X)$ into $\mathscr{M}(X)$ is vaguely continuous. For, let $f \in \mathscr{H}(X)$. One has

where $g(x) = \int f(sx) d\lambda_0(s)$. Now, $g$ is continuous (Ch. VII, §1, No. 1, Lemma 1). On the other hand, let S be the support of $\lambda_0$ and K that of $f$. The conditions $sx \in K$ and $s \in S$ imply $x \in S^{-1}K$; therefore the support of $g$ is contained in $S^{-1}K$, so that $g \in \mathscr{H}(X)$. Then $\langle \lambda_0 * \mu, f \rangle = \langle \mu, g \rangle$ is a vaguely continuous function of $\mu$, which proves our assertion.

2) Let $\mu_0 \in \mathscr{M}(X)$. The mapping $\lambda \mapsto \lambda * \mu_0$ of $\mathscr{C}'(G)$ into $\mathscr{M}(X)$ is continuous for the topologies $\sigma(\mathscr{C}'(G), \mathscr{C}(G))$ and $\sigma(\mathscr{M}(X), \mathscr{H}(X))$. For, let $f \in \mathscr{K}(X)$. Setting $h(s) = \int f(sx)\, d\mu_0(x)$, we have $\langle f, \lambda * \mu_0 \rangle = \langle h, \lambda \rangle$, and $h \in \mathscr{C}(G)$ (Ch. VII, §1, No. 1, Lemma 1).