Proposition 10

Let $n\geqslant 0$ be an integer and let $G_1,\cdots ,G_n$ be locally compact commutative groups. Let $\mu_j$, for $1\leqslant j\leqslant n$, be Haar measures on $G_j$. Let G be the product group of the groups $G_j$ for $1\leqslant j\leqslant n$. Let $\beta$ be the isomorphism of $\widehat{G}$ onto $\prod\widehat{G}_j$ of Prop. 2 of II, p. 206. The Haar measure on $\widehat{G}$ dual to the product Haar measure $\mu=\mu_1\otimes \cdots \otimes \mu_n$ on G identifies with the product of the Haar measures $\widehat{\mu}_j$.

In fact, for every family $(f_j)$ of non-zero elements of $\mathscr{L}^2(G_j)$, the function $f$ on G defined by $(x_j)\mapsto \prod f_j(x_j)$ belong to $\mathscr{L}^2(G)$, and satisfies

by the Plancherel formula, which proves that the product Haar measure of the $\widehat{\mu}_j$ identifies with the Haar measure dual to $\mu$.