Proposition 10

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

Integration, chapter VII, § 1. Construction of a Haar measure

Let $G$ be a locally compact group, $\mu$ a relatively left-invariant measure on $G$ with multiplier $\chi$ . If $\chi_1$ is a continuous representation of $G$ in $\mathbf{C}^*$ , the measure $\chi_1 \cdot \mu$ is relatively left-invariant with multiplier $\chi_1 \chi$ .

For,

\gamma(s)(\chi_1 \cdot \mu) = (\gamma(s)\chi_1) \cdot (\gamma(s)\mu) = (\chi_1(s^{-1})\chi_1) \cdot (\chi(s)^{-1}\mu) = (\chi_1 \chi)(s)^{-1}(\chi_1 \cdot \mu).

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

Cited by

Corollary 1, § 1
Proposition 16, § 1

Cites

Nothing.