Let $(X\_t)\_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.


Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X\_s \in A\}\rvert$
where $\Delta X\_s$ denotes the jump height at time $s$.


For a fixed Borel set $A$ such that $0 \notin A$, let
$N\_t = N(t,A)$.


How can I prove that $E[N\_t] = t\nu(A)$?



Lévy measure and jump behaviour of the corresponding Lévy process
This follows immediately from (say) the following statements in [Schilling - An Introduction to Lévy and Feller Processes](https://arxiv.org/abs/1603.00251):


* parts b) and c) of Lemma 9.4, stating that $N(\cdot,A)$ is a Poisson process of intensity $\nu(A):=EN(1,A)$, where $A$ is any Borel subset of $\mathbb R^d\setminus\{0\}$
* Corollary 9.13, stating that the intensity measure $\nu$ coincides with the Lévy measure.


