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 $$\operatorname P\left[L_t\in B\right]=e^{-\mu(H)t}\sum_{k=0}^\infty\frac{t^k}{k!}\mu^{\ast k}(B)\;\;\;\text{for all }B\in\mathcal B(H)\text{ and }t\ge0.\tag1$$ (We say that $L$ is a compound Poisson process with Lévy measure $\mu$.)
I've stumbled across the definition of the "Poisson random measure" which is "defined" by the formula $$\pi([0,t],B):=\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|\tag2$$ for all $t\ge0$ and $B\in\mathcal B(H\setminus\{0\})$
I'm not familiar with random measures, but couldn't we (more rigorously) treat $\pi$ as the transition kernel with source $(\Omega,\mathcal A)$ and target $([0,\infty)\times(H\setminus\{0\}),\mathcal B([0,\infty))\otimes\mathcal B(H\setminus\{0\}))$ satisfying $$\pi(\omega,[0,t]\times B)=\left|\left\{s\in[0,t]:\Delta L_s(\omega)\in B\right\}\right|\tag3$$ for all $\omega\in\Omega$, $t\ge0$ and $B\in\mathcal B(H\setminus\{0\})$? And what's the point of restricting to $H\setminus\{0\}$?