I do not understand the link between the lines (4) and (5) in the proof of the cross-correlation theorem found in Mathworld:
 (4)
 (5)
(I can't post the images of the equation because I don't have enough reputation)
This confuses me because I thought $\int_{-\infty}^{+\infty}{e^{-2 \pi t} dt}$ was not defined. However the line (5) above implies this quantity is 0.
What don't I understand? Thanks.
From line 4 to 5, the equality $$\int_{-\infty}^\infty e^{2\pi i \tau (\nu'-\nu)}d\tau = \delta(\nu'-\nu)$$ is used, where $\delta$ is the Dirac delta function. The proof given is not rigorous. As you remarked, the function $e^{2\pi i \tau}$ is not in $L^1(-\infty,\infty)$. There are ways to make the equation above (under a suitable interpretation) and the proof itself work rigorously. The step from line 5 to line 6 is also not completely rigorous (or you could say it has gaps).