Theorem 3

Let G be a locally compact group, H a closed subgroup of G, $\mu$ (resp. $\beta$) a left Haar measure on G (resp. H), $\chi$ a continuous representation of G in $\mathbf{C}^*$.

a) In order that there exist on $G/H$ a nonzero measure relatively invariant under G and with multiplier $\chi$, it is necessary and sufficient that $\chi(\xi) = \Delta_H(\xi)/\Delta_G(\xi)$ for all $\xi \in H$.

b) This measure is then unique up to a constant factor; more precisely, it is proportional to $(\chi \cdot \mu)/\beta$.

For there to exist on $G/H$ a nonzero measure relatively invariant under G with multiplier $\chi$, it is necessary and sufficient (Lemma 6) that $\chi \cdot \mu$ be of the form $\lambda^\#$, hence (No. 2, Prop. 4) that $\delta(\xi)(\chi \cdot \mu) = \Delta_H(\xi)(\chi \cdot \mu)$ for all $\xi \in H$. This condition may also be written $\chi(\xi)\chi \cdot \Delta_G(\xi)\mu = \Delta_H(\xi)\chi \cdot \mu$, that is,

for all $\xi \in H$. Whence a). The assertion b) follows at once from Lemma 6 and the fact that the mapping $\lambda \mapsto \lambda^\#$ is injective.

We shall see in §3 (No. 3, Example 4) some very simple examples where the representation $\xi \mapsto \Delta_H(\xi)/\Delta_G(\xi)$ cannot be extended to a continuous representation of G in $\mathbf{C}^*$. In this case, there therefore does not exist any nonzero complex measure on $G/H$ relatively invariant under G.