This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone clarify what is the relationship between the jump measure, levy measure and if it is possible to express these quantities as a $$\sum_{\Delta X \neq 0} sth$$ in the context of Ito's lemma? 

In particular I am looking at the application to the exponential function at the moment, if it simplifies anything.