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?