Corollary 3

Let $n\geqslant 1$ be an integer. The group $\mathbf{R}^n$ is in duality with itself with respect to the mapping $(x, y)\mapsto$ exp(2$i\pi x\cdot y$), and the dual measure of Lebesgue measure on $\mathbf{R}^n$ is Lebesgue measure. The groups $\mathbf{Z}^n$ and $\mathbf{T}^n=\mathbf{R}^n/\mathbf{Z}^n$ are in duality with respect to the mapping obtained by passing to the quotient from the mapping $(n, x)\mapsto$ exp(2$i\pi x\cdot y$), and the dual Haar measure of the counting measure on $\mathbf{Z}^n$ is the normalised Haar measure on $(\mathbf{R}/\mathbf{Z})^n$.

This results from Lemma 7 of II, p. 225, Proposition 10 of II, p. 217 and Corollary 2.