Definition 2

The topological group $\widehat{G}$ is called the dual group of G.

Since G is locally compact, the mapping $(x, \chi )\mapsto \chi (x)$ is continuous on $G\times \widehat{G}$ (TG, X, p. 28, th. 3).

Recall that $\mathscr{M}^1(G)$ denotes the unital involutive Banach algebra of bounded complex measures on G (Example 4 of I, p. 99). For every bounded complex measure $\mu\in \mathscr{M}^1(G)$ and every $\chi \in \widehat{G}$, we write

(cf. INT, VIII, §2, No$^o6$).