Proposition 14

— Let $\alpha$ be a Haar measure on $K^+$. Then the measure $\beta = (\mathrm{mod}_K)^{-1} \cdot \alpha$ on $K^*$ is a left Haar measure on the multiplicative group $K^*$.

For, if $b \in K^*$, the mapping $a \mapsto b^{-1} a$ of K into K transforms $\alpha$ into $(\mathrm{mod}_K b) \alpha$, hence $(\mathrm{mod}_K)^{-1} \cdot \alpha$ into itself, whence the proposition.