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Let $G_1$ and $G_2$ be locally compact abelian groups equipped with Haar measure $\mu_{G_1}$, $\mu_{G_2}$ and $G=G_1 \times G_2$ equipped with $\mu_{G_1} \otimes \mu_{G_2}$.

I would like to show that the associated Haar measure on $\hat{G}=\hat{G_1} \times \hat{G_2}$ is the product $\mu_{\hat{G_1}} \otimes \mu_{\hat{G_2}}$of the associated Haar measures.

Let $f_1\colon G_1\to\mathbb{C}$ and $f_2\colon G_2 \to \mathbb{C}$ be continuous functions with compact support and $f\colon G \to \mathbb{C}$ defined by $(x_1,x_2)\mapsto f_1(x_1)f_2(x_2)$. I proved that $$ \int_{\widehat{G}}|\mathcal{F}f(\chi_1,\chi_2)|^2d\mu_{\widehat{G}}(\chi_1,\chi_2)=\int_{\widehat{G_1} \times \widehat{G_2}} |\mathcal{F}f(\chi_1,\chi_2)|^2 \ d(\mu_{\widehat{G_1}}\otimes \mu_{\widehat{G_2}})(\chi_1,\chi_2). $$

How to complete the proof?

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The essentially thing missing to me was that if a continuous function $f$ with compact support is not identically zero then its Fourier transform is not negligible.

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