quantum field theory - Fourier Transforms Related to Green's Functions

I'm reading a text on field theory where there are a number of assertions made about Fourier transforms that I'm finding confusing. For example, let $G^R = -i \theta(t - t')e^{-i \omega_0 (t - t')}$. The text says that the FT of $G^R$ wtih respect to $t-t'$ is $(\epsilon - \omega_0 + i0)^{-1}$. What is the meaning of this $i0$ term?

Secondly, after inverting this expression the text takes the inverse transform. That is, the inverse FT of $\epsilon - \omega_0 + i0$ and obtains $\delta(t - t')(i\partial_{t'} - \omega_0 + i0)$. What is the meaning of this? How did the fourier transform of an expression become an operator?

Thanks!