Proposition 3

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

Integration, VIII, p. 20, § 3. Convolution of measures on groups

Let $\mu_1, \ldots, \mu_n$ be measures on G. If all of the $\mu_i$ , except at most one, have compact support, then the $\mu_i$ are convolvable.

For, let $S_i$ be the support of $\mu_i$ , and suppose that $S_i$ is compact for $i \neq i_0$ . Let K be a compact subset of G. The set of $(x_1, \ldots, x_n) \in \prod_i S_i$ such that $x_1 x_2 \cdots x_n \in K$ is compact, because the conditions $x_i \in S_i$ for all $i$ , $x_1 x_2 \cdots x_n \in K$ imply

x_{i_0} \in S_{i_0-1}^{-1} \cdots S_1^{-1} K S_n^{-1} \cdots S_{i_0+1}^{-1} .

Therefore the $\mu_i$ are convolvable (§1, No. 4, Prop. 4).

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

Cited by

Nothing.

Cites

§1, No. 4, Prop. 4