Lemma 2

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

Integration, VII, p. 56, § 3. Applications and examples

Let $G$ be a compact group, $\mu$ the normalized Haar measure of $G$ , $\mathbf{E}$ a finite-dimensional real affine space, $A$ the affine group of $\mathbf{E}$ , and $\rho$ a homomorphism of $G$ into $A$ . Assume that, for every $x \in \mathbf{E}$ , the mapping $s \mapsto \rho(s)x$ of $G$ into $\mathbf{E}$ is continuous. Then, for every $x \in \mathbf{E}$ , the point

x_0 = \int_G \rho(s)x\, d\mu(s) \in \mathbf{E}

is invariant under $G$ .

For, for every $t \in G$ ,

\rho(t)x_0 = \int_G \rho(t)\rho(s)x\, d\mu(s) = \int_G \rho(ts)x\, d\mu(s) = \int_G \rho(s)x\, d\mu(s) = x_0.

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

Cited by

Exercise 12, § 3
Example 8, § 3
Proposition 3, § 3

Cites

Nothing.