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I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither author delves into any detail about discontinuous characters on locally compact abelian groups (or even gives mention of an example). Clearly such characters exist else we would not look specifically at continuous characters when discussing the dual group.

I tried to come up with an example in the case of $G = (\mathbb R,+)$ for the last hour or so but have gotten nowhere. I considered trying to define the character to be $1$ on dyadic numbers and $-1$ on the complement; I tried looking at an irrational translation of the rationals and letting the character be $1$ on this and $-1$ on the complement; and I even looking at $\mathbb R$ as a vector space over $\mathbb Q$ and trying to define characters via this approach somehow but things got hopelessly complicated at this stage.

Unsurprisingly, got absolutely nowhere in any of these. I get the feeling that the last one is probably relatively close to the kind of methodology one would need to define discontinuous characters (I'm guessing there are some pretty subtle set-theoretic undertones to such a problem given how challenging this has been). Can anyone shed light on this problem in any way? Can such a character be constructed or an algorithm/program be given to find one?

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You need to use the axiom of choice or something like that: http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation - and your comment about vector spaces over $\mathbb Q$ is exactly the right way to go.

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  • $\begingroup$ Does the same argument hold in this case? We're interested in a multiplicative not an additive homomorphism. $\endgroup$ – Cameron Williams Jun 4 '14 at 2:26
  • $\begingroup$ You can get to multiplicative homomorphisms by considering $x \mapsto e^{f(x)}$. If you want the homomorphism to be bounded, use $x \mapsto e^{i f(x)}$. $\endgroup$ – Stephen Montgomery-Smith Jun 4 '14 at 2:28
  • $\begingroup$ Good point! Well that's a big let down haha. I was hoping it was not going to be quite so grueling a construction. $\endgroup$ – Cameron Williams Jun 4 '14 at 2:31
  • $\begingroup$ Yes, but you can take solace in that you thought of a question so interesting that it has occupied some of the great minds of mathematics, $\endgroup$ – Stephen Montgomery-Smith Jun 4 '14 at 2:32
  • $\begingroup$ That does make me feel better. An hour wasted hardly seems significant compared to the headaches I'm sure it's caused many other people. Thanks for the link! I have seen that webpage several times here on MSE but it seems to have completely slipped my mind. $\endgroup$ – Cameron Williams Jun 4 '14 at 2:34

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