How to build a Poisson random measure according a Levy Process

Question

Let $(\Omega, \mathcal{F}, P)$ and $(\Theta, \mathcal{B}, \rho)$. A Poisson random measure (PRM) with intensity $\rho$ is a kernel $\mathcal{N}: \Omega \times \mathcal{B} \to \mathbb{R}$ such that:

1. $\forall \omega \in \Omega$, $\mathcal{N}(\omega, \cdot)$ is a measure on $(\Theta, \mathcal{B})$

2. $\forall B \in \mathcal{B}$, $\mathcal{N}( \cdot, B) \sim Poisson(\rho (B))$

3. If $B_1,...,B_k$ are disjoint, then $N( \cdot, B_1), ..., N( \cdot, B_k)$ are independent (Poisson) random variables

First, Sato (Lévy Processes and Infinitely Divisible Distributions ) shows that there exists a PRM (page 122) :

PROPOSITION 19.4. For any given $\sigma$-finite measure space $(\theta, \mathcal{B}, \rho)$, there exists, on some probability space $\left(\Omega^{0}, \mathcal{F}^{0}, P^{0}\right)$, a Poisson random measure $\{N(B): B \in \mathcal{B}\}$ on $\Theta$ with intensity measure $\rho$.

The idea is to take a sequence $\left\{Z_{n}: n=1,2, \ldots\right\}$ of independent identically distributed random variables defined in $(\Omega, \mathcal{F}, P)$ with values on $\Theta$ each with distribution $\rho(\Theta)^{-1} \rho$ and a Poisson random variable $Y$ with mean $\rho(\Theta)$ such that $Y$ and $\left\{Z_{n}\right\}$ are independent. Define $\mathcal{N}( \omega , B)=0$ if $Y=0$, and

\begin{equation}\tag{1} \mathcal{N}( \omega , B)=\sum_{j=1}^{Y} \mathbf{1}_{B}\left(Z_{j}(\omega)\right) \end{equation}

if $Y \geq 1$.

So I am trying to adapt this for the case of a Levy Process.

Given a Levy Process $X = [X_t : t \geq 0] \sim (b,\sigma, \nu)$ (Lévy–Khintchine representation). Take $\Theta = (0,\infty) \times (\mathbb{R}^d\setminus \{0\})$. Define $\rho( (0,t) \times A ) = t \nu(A)$, $\, \forall \, ( (0,t) \times A )\in \Theta = (0,\infty) \times (\mathbb{R}^d\setminus \{0\})$. Now, define

\begin{equation}\tag{2} \mathcal{N} ( \omega, (0,t) \times A ) = \#\left\{ s \in (0,t) : \Delta X_{s}(\omega) \in A \right\} \end{equation}

So my question is

1. how to do Eq (2) equivalent to Eq (1);

2. how to show that $\mathcal{N} ( \cdot , (0,t) \times A ) \sim Poisson ( \rho( (0,t) \times A ) ) = Poisson (t \nu(A))$

My attempt:

My attempt is try to define, for all $A \subset \mathbb{R}^d$ and $\omega \in \Omega$,: $$Z_1^A = \inf \{ s>0 : \Delta X_s(\omega) \in A \} $$ $$Z_n^A = \inf \{ s> Z_{n-1}^A : \Delta X_s(\omega) \in A \} $$

Thus \begin{equation} \mathcal{N}( \omega , B= (0,t) \times A)=\sum_{j=1}^{\infty} \mathbf{1}_{(0,t)}\left(Z_{j}^A(\omega)\right)=\sum_{j=1}^{\infty} \mathbf{1}_{\{ Z_{j}^A(\omega) < t \} } \end{equation} This is as close as I can get to equation (1), but I don't know how to proceed or prove questions 1 and 2.