The Levy-Khintchine formula gives a triple $(a,\sigma,\nu)$ for the characteristic exponent $\Psi(s)$ of an infinitely divisible random variable where
$\Psi(s)=ias + \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} + is\mathbb{1}_{|x|<1})d\nu(x)$
My question is whether $\nu$ is unique? Or whether it is only the unique Levy measure?
Are there references in the literature to this fact.
Thanks