Lemma 6

Let $\lambda$ be a measure on $G/H$, $\chi$ a continuous representation of G in $\mathbf{C}^*$. The following properties are equivalent: a) $\lambda$ is relatively invariant on $G/H$ with multiplier $\chi$; b) $\lambda^\#$ is relatively invariant on G with left multiplier $\chi$; c) $\lambda^\#$ is of the form $a \chi \cdot \mu$ ($a \in \mathbf{C}$).

The condition a) means that, for every $s \in G$,

this is equivalent to $(\gamma_{G/H}(s)\lambda)^\# = \chi(s)^{-1}\lambda^\#$, that is, to

Whence the equivalence of a) and b). The equivalence of b) and c) follows from §1, No. 8, Cor. 1 of Prop. 10.