The Lévy and jump measures are two key characteristics of Lévy processes. This paper fills what seems to be a simple gap in the literature, by giving an explicit relation between the jump measure, which is a Poisson random measure, and the L´evy measure. This relation paves the way to a simple proof of the classical result on path continuity of Lévy processes in Section 2.
The jump function in Paul Lévy’s version of the Lévy-Khinchine formula and the Lévy measure in more recent characterizations essentially play the same role, but with different drift and Gaussian components. This point is shown in detail in Section 3, together with an explicit relation between the jump function and the Lévy measure.