# Lévy measure and jump behaviour of the corresponding Lévy process

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 $$E[N_t] = t\nu(A)$$?

• Please don't post content where every single formula is an image. MathOverflow supports MathJax, and that is what you should be using. I have edited accordingly. What is the relationship between $A$ and $B$? Jan 26 at 18:23
• Sorry, I don't know how to use Mathoverflow. The borel set B is A, I'm confused. Jan 26 at 18:28

• 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.