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Let $G$ be a topological group. Let's assume $G$ is abelian, locally compact and Hausdorff. Then there exists is a Haar measure $d \nu$ for $G$ and $d \mu$ for $\hat G$. Under the assumption that the total variation of $d \mu$ is finite we can define

$$x \mapsto \int_{\hat G} \chi(x) d\mu(\chi)$$

as a function from $G$ to $\mathbb C$. Can we show that this defines a continuous function?

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