# The Lévy process jumps

I have two questions.

Let $$(X_t)_{t\geq 0}$$ be a Lévy process with Lévy measure $$\nu$$. The jump process $$\Delta X=\left(\Delta X_t\right)_{t\geq 0}$$ is defined by

$$\Delta X_t=X_t-X_{t-}$$, for every $$t\geq0$$, with $$X_{t-}$$ left limit in $$t$$.

For every $$0\leq t <\infty$$ e $$A \in \mathcal{B}(\mathcal{R}-\{0\})$$, let

$$N(t,A)(w)= \#\left\{ 0 \leq s \leq t: \Delta X_t(w) \in A \right\}$$ if $$w \in \Omega_0$$

and

$$N(t,A)(w)=0$$ if $$w \in \Omega_0^c$$, with $$\Omega_0$$ a measurable set with probability $$1$$ such that $$t\longrightarrow X_t(w)$$ is cadlag for every $$w \in \Omega_0$$.

Let $$\nu(\cdot)=\mathbb{E}\left[N(1,\cdot)\right]$$ be the intensity measure. We say that $$A$$ is bounded below if $$0 \notin \overline{A}$$.

Theorem

1.If $$A$$ is bounded below, then $$\left(N(t,A)\right)_{t\geq 0}$$ is a Poisson process of intensity $$\nu(A)$$.

2.If $$A_1, \dots , A_m \in \mathcal{B}(\mathcal{R}-\{0\})$$ are disjoint e bounded below then the random variable $$N(t,A_1) \dots N(t,A_m)$$ are indpendent.

So my questions are:

i) How can I prove the second statement?

ii) How can I deduce from this theorem that $$\nu(A)<\infty$$ for all $$A$$ bounded below?