Question about interchanging series and integral How can I reason the following interchange of integral and series:
$$\int_{-\infty}^{\infty}\lim_{N\to \infty} \sum_{n=0}^{N}\frac{1}{2^n}1_{[2n-2,2n+2]}(t)e^{-itx}dt=\lim_{N\to \infty}\sum_{n=0}^{N}\int_{-\infty}^{\infty}\frac{1}{2^n}1_{[2n-2,2n+2]}(t)e^{-itx}dt$$
Because $e^{-itx}$ can get smaller than $0$, I can't use Fubini.

You can interchange because:
$$\sum_{n=0}^{\infty}\int\frac{1}{2^n}1_{[2n-2,2n+2]}(t)dt=\sum_{n=0}^{\infty}\frac{4}{2^n}< \infty$$
So from $DCT$ you have the desired conclusion.
