Corollary 1

Let $\mu$ be a left Haar measure on $G$. For a nonzero measure $\nu$ on $G$ to be relatively left-invariant, it is necessary and sufficient that it be of the form $a \chi \cdot \mu$, where $a \in \mathbf{C}^*$ and $\chi$ is a continuous representation of $G$ in $\mathbf{C}^*$; its multiplier is then $\chi$.

The condition is sufficient (Prop. 10). On the other hand, if $\nu$ is a nonzero relatively left-invariant measure with multiplier $\chi$, then $\chi^{-1} \cdot \nu$ is left-invariant (Prop. 10) hence is of the form $a \mu$ with $a \in \mathbf{C}^*$ (No. 2, Cor. of Th. 1).