# Existence of jump intensity (random) measure of a Lévy process

Let $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space, $$E$$ be a normed $$\mathbb R$$-vector space and $$(X_t)_{t\ge0}$$ be an $$E$$-valued càdlàg Lèvy process on $$(\Omega,\mathcal A,\operatorname P)$$.

How can we prove that there is a (unique) transition kernel $$\pi$$ from $$(\Omega,\mathcal A)$$ to $$([0,\infty)\times E,\mathcal B([0,\infty)\times E))$$ with$$^1$$ $$\pi(\omega,[0,t]\times B\setminus\{0\})=|\{s\in[0,t]:\Delta X_s(\omega)\}\|\tag1$$ for all $$\omega\in\Omega$$, $$t\ge0$$ and $$B\in\mathcal B(E)$$?

Clearly, for fixed $$\omega\in\Omega$$, we should be able to find a unique measure $$\pi_\omega$$ on $$([0,\infty)\times E,\mathcal B([0,\infty)\times E))$$ with $$\pi_\omega([0,t]\times B\setminus\{0\})=|\{s\in[0,t]:\Delta X_s(\omega)\}\|\tag2$$ for all $$t\ge0$$ and $$B\in\mathcal B(E)$$ by the classical measure-theoretic results, since $$\mathcal B([0,\infty))$$ is generated by the $$\cap$$-stable system $$\{[0,t]:t\ge0\}$$. The only thing I'm unsure about is whether we need to specify the values for $$\{0\}\subseteq E$$, since $$\mathcal B(E)$$ is clearly not generated by $$\{B\setminus\{0\}:B\in\mathcal B(E)\}$$ to obtain the uniqueness.

However, even when the latter issue is solved, I'm not sure how we obtain the $$\mathcal A$$-measurability of $$\Omega\ni\omega\mapsto\pi_\omega(A)$$ for every $$A\in\mathcal B([0,\infty)\times E)$$ ...

As usual, $$x(t-):=\lim_{s\to t-}$$ and $$\Delta x(t):=x(t)-x(t-)$$ for $$t\ge0$$, where $$x(0-):=x(0)$$.