Theorem 2

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

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

*Let $G$ be a locally compact group, $H$ a closed subgroup of $G$ , $\mu$ a left Haar measure on $G$ , and $\beta$ a left Haar measure on $H$ .

a) There exist functions $\rho$ continuous and $> 0$ on $G$ , such that

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

for all $x \in G$ and $\xi \in H$ .

b) Given such a function $\rho$ , one can form the measure $\lambda = (\rho \cdot \mu)/\beta$ on $G/H$ , and $\lambda$ is a nonzero positive measure quasi-invariant under $G$ .

c) For $s, x$ in $G$ , $\rho(sx)/\rho(x)$ depends only on $s$ and $\pi(x)$ , hence defines a function $\chi$ continuous and $> 0$ on $G \times (G/H)$ such that

\chi(s, \pi(x)) = \frac{\rho(sx)}{\rho(x)}.

Then

\gamma_{G/H}(s)\lambda = \chi(s^{-1}, .) \cdot \lambda \quad \text{for all } s \in G.

a) follows from Prop. 7. b) follows from Lemmas 5 and 4. c) follows from (17).

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

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Cites

Prop. 7