Proposition 16

— Let $\alpha$ be a Haar measure of the additive group of $A$. The measures

on $A^*$ are, respectively, left and right Haar measures of the multiplicative group $A^*$.

Let $\alpha'$ be the restriction of $\alpha$ to the open set $A^*$. For $a \in A^*$, one has $L_a(\alpha') = (\mathrm{mod}_K N_{A/K}(a))^{-1} \alpha'$, therefore $(\mathrm{mod}_K N_{A/K}(a))^{-1} d\alpha'(a)$ is a left Haar measure on $A^*$ (No. 8, Cor. 1 of Prop. 10). Passing to the opposite algebra, one sees that $(\mathrm{mod}_K N_{A^0/K}(a))^{-1} d\alpha'(a)$ is a right Haar measure on $A^*$.