Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$L_t=\sum_{i=1}^{N_t}Y_i\;\;\;\text{for all }t\ge0\tag1$$ for some $H$-valued independent identically distributed process $(Y_n)_{n\in\mathbb N}$ on $(\Omega,\mathcal A,\operatorname P)$ with $Y_1\sim\lambda^{-1}\mu$ for some $\lambda>0$ and some càdlàg Poisson process $(N_t)_{t\ge0}$ on $(\Omega,\mathcal A,\operatorname P)$.
Define $$\pi(\omega,A\times B):=\left|\left\{t\in A:\Delta L_t(\omega)\in B\right\}\right|$$ for $\omega\in\Omega$ and $(A,B)\in\mathcal B([0,\infty))\times\mathcal B(H).$$
Given $B\in\mathcal B(H)$, what's the easiest way to see that $$X_t(\omega):=\pi(\omega,[0,t]\times B)\;\;\;\text{for }(\omega,t)\in\Omega\times[0,\infty)$$ is a Poisson process on $(\Omega,\mathcal A,\operatorname P)$ with intensity $\mu(B)$?
To get a start, it would even help if someone could show how we can derive a formula for $\operatorname P\left[X_s=k,X_t-X_s=l\right]$, where $k,l\in\mathbb N$ and $t\ge s\ge0$, from which we could conclude that $X_s$ and $X_t-X_s$ are independent and have the desired distributions.