Lemma 8

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

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

— Let $F$ be a fundamental domain. For every $x \in X$ ,

\sum_{s \in H} \varphi_{Fs}(x) = n(x).

(24)

Since $\varphi_{Fs}(xt) = \varphi_{Fst^{-1}}(x)$ for all $s$ and $t$ in $H$ , the two members of (24) remain invariant when $x$ is replaced by $xt$ . We can therefore suppose that $x \in F$ . We then have the equivalences

\varphi_{Fs}(x) = 1 \iff x \in Fs \iff xs^{-1} \in F \iff xs^{-1} = x \iff s \in H_x,

whence (24).

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

Cited by

Proposition 15, § 2
Theorem 4, § 2

Cites

Nothing.