In my current work, I encountered a problem that I need to find the integral of a summation of complex functions: Let $f(\omega)$ and $g(\omega)$ be finite-valued complex functions (which are the Fourier transform of real functions), I want to find the integral: $$\int_{-h}^h\left\{\sum_{u=-U}^U\sum_{v=-V}^V\sum_{\omega=-M}^{M}f_u(\omega)g_v(\omega)e^{i\omega(u+v)r}\right\}\mathrm dr, r \in \mathbb{R}$$
I wonder whether I can rewrite the function as follows: $$\sum_{u=-U}^U\sum_{v=-V}^V\sum_{\omega=-M}^{M}\left\{f_u(\omega)g_v(\omega)\int_{-h}^he^{i\omega(u+v)r}dr\right\}$$
I think Fubini/Tonelli's theorem may help but I am not sure because $f$, $g$, and $e^{i\omega^T(u+v)r}$ are complex here.
You can rewrite your expression in the intended way. This is not a matter of Fubini's theorem but just the linearity of the integral, valid for real-valued, complex-valued, or vector-valued functions: $$\int_a^b\bigl(\lambda f(x)+\mu g(x)\bigr)\>dx=\lambda\int_a^b f(x)\>dx+\mu\int_a^b g(x)\>dx\ .$$ Fubini's theorem is about multivariate integrals, while in your case we just have the integration $\int_{-h}^h \ldots dr$.