Proposition 12

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

If $\lambda \in \mathscr{C}'(G)$ and $\mu \in \mathscr{M}(X)$ , then

\gamma(\lambda)\mu = \lambda * \mu.

By Prop. 7 of §1, No. 5,

\lambda * \mu = \int_G (\varepsilon_s * \mu) d\lambda(s).

Now, $\varepsilon_s * \mu = \gamma(s)\mu$ (No. 2, formula (5)) and

\int_G (\gamma(s)\mu) d\lambda(s) = \gamma(\lambda)\mu

by the definition of $\gamma(\lambda)$ .

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

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