Remark 2

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

Integration, VII, p. 36, § 2. Quotient of a space by a group; homogeneous spaces

Suppose $H$ is compact and $\beta$ is normalized. The saturation of every compact subset of $X$ is compact. Therefore, if $f \in \mathscr{K}(X/H)$ then $f \circ \pi \in \mathscr{K}(X)$ ; and, for every positive measure $\lambda$ on $X/H$ , Prop. 5 c) gives

\int_X (f \circ \pi)(x)\, d\lambda^\#(x) = \int_{X/H} f(u)\, d\lambda(u).

In other words, $\lambda$ is the image of $\lambda^\#$ under $\pi$ .

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

Cited by

Proposition 13, § 2

Cites

Prop. 5