Lemma 5

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

Integration, VII, p. 41, § 2. Quotient of a space by a group; homogeneous spaces

Let $\mu$ be a left Haar measure on $G$ , and $\rho$ a locally $\mu$ -integrable function. For $\rho \cdot \mu$ to be of the form $\lambda^\#$ , it is necessary and sufficient that, for every $\xi \in H$ ,

\rho(x\xi) = \frac{\Delta_H(\xi)}{\Delta_G(\xi)} \rho(x)

locally $\mu$ -almost everywhere on $G$ .

To say that $\rho \cdot \mu$ is of the form $\lambda^\#$ amounts to saying that, for every $\xi \in H$ , $\delta(\xi)(\rho \cdot \mu) = \Delta_H(\xi)\rho \cdot \mu$ (Prop. 4). Now,

\delta(\xi)(\rho \cdot \mu) = (\delta(\xi)\rho) \cdot (\delta(\xi)\mu) = \Delta_G(\xi)(\delta(\xi)\rho) \cdot \mu,

whence the lemma.

We can now establish the announced existence result, and even a more precise result:

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

Cited by

Nothing.

Cites

Prop. 4