Is this equation with integrals true? If it is, how?

Question

This equation is from the physics book. It says we rearrange integrals in left part and get the right part, but I don't understand how it is done. $$ \frac{1}{2\pi}\int_{-\infty}^{\infty}dt \left ( \int_{-\infty}^{\infty}d\omega'C\left ( \omega' \right ) e^{-i \omega' t} \right ) e^{i \omega t} = \frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega'C\left ( \omega' \right )\int_{-\infty}^{\infty}dt e^{i\left ( \omega - \omega' \right ) t} $$

Answer 1

This is just a change in order of integration, from first going from $\omega'$ to $t,$ in reverse order. And your domain is a rectangle, albeit an infinite one.

Answer 2

This change of the order of integration operators is analogous to $\sum_i\sum_ja_{ij}=\sum_j\sum_ia_{ij}$. Conditions sufficient for this change to be legal are given in Fubini's theorem.