Proposition 15

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

Integration, VII, p. 51, § 2. Quotient of a space by a group; homogeneous spaces

Assume that $X$ is countable at infinity. Let $\mu$ be a measure $\geq 0$ on $X$ . Let $F$ be a fundamental domain such that $Fs$ is $\mu$ -measurable for every $s \in H$ . Let $f$ be a $\mu$ -integrable function on $X$ , with values in a Banach space or in $\overline{\mathbf{R}}$ . Then the family of the

\int_{Fs} n(x)^{-1} f(x) d\mu(x) \qquad (s \in H)

is summable, and

\int_X f(x) d\mu(x) = \sum_{s \in H} \int_{Fs} n(x)^{-1} f(x) d\mu(x).

If $A$ is a finite subset of $H$ , then

\left| \sum_{s \in A} n^{-1} f \varphi_{Fs} \right| \leq n^{-1} |f| \sum_{s \in A} \varphi_{Fs} \leq |f|

by Lemma 8. Lemma 8 also proves that

\sum_{s \in A} n^{-1} f \varphi_{Fs}

converges pointwise to

f

with respect to the increasing directed set of finite subsets of

H

. Prop. 15 then follows from Ch. IV, §4, No. 3, Th. 2.

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

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Exercise 28, § 1

Proposition 15

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