Proposition 10

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

Integration, chapter VII, § 2. Quotient of a space by a group; homogeneous spaces

— Let $G$ be a locally compact group, $G'$ a closed normal subgroup, $G''$ the group $G/G'$ , $\pi$ the canonical mapping of $G$ onto $G/G'$ , and $\alpha, \alpha', \alpha''$ left Haar measures on $G, G', G''$ .

a) Multiplying $\alpha$ by a constant factor if necessary, we can suppose that $\alpha'' = \alpha/\alpha'$ . In particular, if $f \in \mathscr{K}(G)$ then

\int_G f(x)\,d\alpha(x) = \int_{G''} d\alpha''(\dot{x}) \int_{G'} f(x\xi)\,d\alpha'(\xi) \quad (\dot{x} = \pi(x)).

b) One has $\Delta_G(\xi) = \Delta_{G'}(\xi)$ for all $\xi \in G'$ ; in particular, if $G$ is unimodular then so is $G'$ .

c) The kernel of the representation $\Delta_G$ of $G$ in $\mathbf{R}_+^*$ is the largest unimodular closed normal subgroup of $G$ .

Applying Th. 3 of No. 6 with $\chi = 1$ (and in the knowledge that here, there exists a measure on $G/G'$ invariant under $G$ , namely $\alpha''$ ), we obtain a) and b); c) follows at once from b).

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

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Cites

Th. 3