I have never seen the proof that the number of jumps in a continuous time Markov chain is a random varaible. We assume that the process have right-continuous paths. Do you know the proof?
What I am able to do is prove that the event that there is a jump at a particular time $t$ is measurable event. I do it like this:
Assume that the process is $S_t$ is defined on a probability space $(\Omega,\mathcal{A},P)$, we assume that it is right-continuous. We also have a countable state space $\mathbb{S}$. Then the event that there is a jump at time $t$ is
$\bigcup\limits_{s\in\mathbb{S}}\left(S_t^{-1}(\{s\})\bigcap\limits_{n \in \mathbb{N}}\left[\bigcup\limits_{q\in(t-1/n,t)\cap\mathbb{Q}}S_t^{-1}(\mathbb{S}\backslash\{s\} )\right] \right)$.
But this is just at a given time $t$. What I am having trouble with is adding all these events up for lets say an interval. Do you see how to do this? Are the number of jumps in a continuous time markov chain a random variable?