Suppose $(\Omega, \mathcal{F}, P)$ is a probability space. Assume $(X_{t}, P)$ is a Levy process with generating triplet $( 0, 0, \nu)$ with $X_{0}=0$. This means there is no continuous part in $X_{t}$. My question, what kinds of condition on the Levy measure $\nu$ can lead to the following identity: for any $s>0$,
$ P(\omega: X_{s-}(\omega)=0)=0 $.
My guess is that $\nu$ is absolute continuous with respect to Lebesgue measure on real line and with non-zero radon-nikodym density. This question is related to (https://mathoverflow.net/questions/159811/the-probability-of-levy-process-staying-at-a-point)
Any reference is appreciated.