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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.