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I'm a bit confused - How is the Fourier transform of a measure on a compact abelian group defined? specifically the Fourier transform of a measure on $\mathbb{T}$ the unit circle in the complex plain.

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    This is developed quite thoroughly at http://terrytao.wordpress.com/2009/04/06/the-fourier-transform/, for example. Google is helpful with such an inquiry.2012-08-08

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If $\mu$ is a measure on the compact abelian group $G$ and $\gamma$ is in the dual group, $\hat{\mu}(\gamma) = \int_G (-g, \gamma)\ d\mu(g)$ In the case ${\mathbb T}$, the dual group is $\mathbb Z$, acting on $\mathbb T$ by $(n, \omega) = \omega^n$.