Corollary

a) The group $\mathbf{Q}_p$ is in duality with itself with respect to the mapping $(x, y)\mapsto$ exp(2$i\pi \lambda (xy)$). The normalised Haar measure on $\mathbf{Q}_p$ is then its own dual ;

b) The groups $\mathbf{Z}_p$ and $\mathbf{Q}_p/\mathbf{Z}_p$ are in duality relative to the mapping obtained by passing to the quotient from the mapping defined by $(z, x)\mapsto$ exp(2$i\pi \lambda (zx)$), and the dual measure of the normalized Haar measure on $\mathbf{Z}_p$ is the counting measure on $\mathbf{Q}_p/\mathbf{Z}_p$.

The proof follows step by step that of Cor. 2 of Prop. 19.