Fourier transform of a measure

Question

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.

Answer 1

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 $(n, \omega) = \omega^n$.