Let $G$ be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on $\hat G$
$$f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) \mu_G(x)$$
where $\Psi$ is the kernel of the Fourier transform (in case $G=\Bbb R$ it is just $e^{2\pi i x \xi}$ for example), and $\mu_G$ is an invariant Haar measure on $G$.
For what class of functions do we have the Fourier transform? Certainly for almost everywhere continuous and compactly supported, but in $\Bbb R$ we can relax the last condition to rapid decay at infinity (as in functions the Schwartz space).
What does $\Psi\in \mathcal F(G \times \hat G)$ look like for other LCA? Is it just the pairing of $G$ with $\hat G$? (n.b. I don't really know what I mean by $\mathcal F$, is it the same space where the Fourier transform is defined?)
Let $\mu'$ denote the measure on the dual group so that the Fourier inversion formula given by $$\hat f\mapsto f(x) = \int_{\hat G}\hat f(\xi)\overline{\Psi(x,\xi)}\mu'_{\hat G}(\xi)$$ works. How do I prove that $(\mu')'=\mu$? I have been trying to write down a formula for $\mu'$ in terms of $\mu$ but I got lost in my integrals.