Proposition 14

For every bounded complex measure $\mu$ on $M$, the function $\mathscr{L}\mu$ on $X$ is continuous and bounded. If $\mathscr{M}_+^b(M)$ is equipped with the tight topology and $\mathscr{C}^b(X; \mathbf{C})$ with the topology of compact convergence, the mapping $\mu \mapsto \mathscr{L}\mu$ of $\mathscr{M}_+^b(M)$ into $\mathscr{C}^b(X; \mathbf{C})$ is continuous.

The set of characters of $M$ that tend to 0 at infinity will be denoted $X_0$; this set is stable under multiplication. We shall say that a submonoid$^{(1)}$ S of $X$ is full if S is stable for the mapping $\chi \mapsto \overline{\chi}$, $S \cap X_0$ separates the points of $M$ (GT, X, §4, No. 1, Def. 1) and if, given any $m \in M$, there exists an element $\chi$ of $S \cap X_0$ such that $\chi(m) \neq 0$.

Suppose in addition that $M$ is a noncompact abelian group. Let $f$ be a function on $M$ that tends to 0 at infinity; the same is then true of the function $x \mapsto f(x)f(-x)$ on $M$, whereas every character $\chi$ of $M$ satisfies $\chi(x)\chi(-x) = \chi(0) = 1$. It follows that $X_0$ is empty, and that $X$ does not contain any full submonoid. Thus, Theorem 3 below does not apply to locally compact groups that are not compact.