A Borel measure $\nu$ on $\mathbb{R}$ is called a Lévy measure if $\nu({0})=0$ and $\int_\mathbb{R}(1\wedge|x^2|) \, \nu(dx) < \infty .$ (https://en.wikipedia.org/wiki/Financial_models_with_long-tailed_distributions_and_volatility_clustering#Infinitely_divisible_distributions)
So, what exactly is $(1\wedge|x^2|)$? (Or rather correctly, what is the definition of levy measure saying?)
Edit: OK, but then why is levy measure needed?
$$1\wedge|x^2|=\begin{cases}1 & \text{if} & |x|\gt1, \\ x^2 & \text{if} & |x|\leqslant1. \end{cases}$$