In the Lévy-Itô decomposition it's necessary to compensate small jumps. That's clear. The small jumps are perhaps non-summable. But why are the jumps quared summalbe?
In the "ordinary" proofs of the Lévy Khintchine Formula or the Lévy-Itô Decomposition i can't get the point where we exactly use THIS and where the proof fails if we would not use this. I think there is a connection in both proofs, because the integrand in the L-K formula is near the origin something to the power of 2.
Are there basic results where it's proven that the x^2 integrated with respect to a Lévy measure is finite without using the Lévy-Khintchine Formula or the L-I decomposition?