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?



The Lévy process jumps
Your questions are covered by, say, Theorem 3 in Section 4 of [Lalley's notes](http://galton.uchicago.edu/%7Elalley/Courses/385/LevyProcesses10-23-2017.pdf). The proof (given there) is somewhat involved.


