Proposition 4

Tag 09F3, permanent. Cite it as [Bourbaki, Tag 09F3].

Integration, VII, p. 31, § 2. Quotient of a space by a group; homogeneous spaces

a) Let $\lambda$ be a measure on $X/H$ . There exists one and only one measure $\lambda^\#$ on $X$ such that

\int_{X/H} f^b d\lambda = \int_X f d\lambda^\#

for all $f \in \mathscr{K}(X)$ . One has $\delta(\xi)\lambda^\# = \Delta_H(\xi)\lambda^\#$ for all $\xi \in H$ .

b) Conversely, let $\mu$ be a measure on $X$ such that $\delta(\xi)\mu = \Delta_H(\xi)\mu$ for all $\xi \in H$ . There exists one and only one measure $\lambda$ on $X/H$ such that $\mu = \lambda^\#$ .

This is a special case of No. 1, Prop. 3.

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

Cited by

§ 2. Quotient of a space by a group; homogeneous spaces
Definition 1, § 2
Lemma 5, § 2
Corollary 2, § 2
Theorem 3, § 2

Cites

No. 1, Prop. 3