Proposition 7

Tag 09JH, permanent. Cite it as [Bourbaki, Tag 09JH].

Integration, VIII, p. 21, § 3. Convolution of measures on groups

*Let G and G' be two locally compact groups, u a continuous representation of G in G', and $\mu_1, \ldots, \mu_n$ measures on G, all $\neq 0$ . Then the following assertions are equivalent: (i) $u$ is $\mu_i$ -proper for all i, and the measures $u(|\mu_i|)$ are convolvable; (ii) the $\mu_i$ are convolvable and $u$ is proper for $*_{i}(|\mu_i|)$ . When these conditions are satisfied,

*_{i} u(\mu_i) = u(*_{i} \mu_i).

This follows from §1, No. 2, Cor. of Prop. 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 3

Cites

§1, No. 2, Cor. of Prop. 1