While studying Lévy processes and Poisson processes, I have encountered the jump random measure $$\mu^{X}(\omega, \cdot)=\sum_{t \geq 0} \delta_{\left(t, \Delta X_{t}(\omega)\right)} .$$ My confusion lies within the definition of random measure, since many textbooks introduce a random measure differently.
Kallenberg (Probability and its Applications) introduces it as
By a random measure on $S$ we mean a ($\sigma$-finite) kernel from the basic probability space $(\Omega, \mathcal A, P)$ into $\mathcal S$.
With (transition) kernel we mean as in https://en.wikipedia.org/wiki/Transition_kernel, so we want it to be a measure when fixing $\omega\in\Omega$, and want it to be a random variable when fixing a Borel set $A\in\mathcal B(\mathbb R_+\times \mathbb R\backslash \{0\}).$
However, in Appelbaum (Lévy Processes and Stochastic Calculus) on p. 103, it defines a random measure without the assumption it is a measure when fixing $\omega$. And, on the other hand, in Jacod and Shiryaev (Limit theorems for stochastic processes) on p. 65, a random measure is a family of random measure, but they do not state the condition we want it to be a random variable when fixing $A$.
Any comments that would resolve my confusion? I am trying to understand why $\mu^X$ is well-defined.