This is a proposition from my lecture notes (no proof is given -- it's left as an exercise). The section on Poisson random measures starts on page 43 of these lecture notes. At this point in the course, we hadn't got to Lévy processes -- they're chapter nine -- but I now known that "cadlag with stationary, independent increments" means "Lévy" (basically); however, of course, I don't want to use general Lévy process results. There is a very similar claim in Theorem 8.2.1. I expect to have a proof reasonably similar to this, but I haven't been able to get anywhere. In the proof of Theorem 8.2.1, it claims a certain form of the Poisson random measure $M$. I don't 100% see why this is allowed, and thus can't see if I can apply it in this situation. (I mean, I could just assume that it still works, pretend it's applied maths and hope for the best!)
I don't really have a very good understanding about PRMs in general. For example, I'm really not sure how to take a condition expectation when the random thing is a measure. I mean, I could just push through some definitions, like the proof of T8.2.1, but I'd be interested in getting a better understanding about them in general. If anyone could offer me some insight/intuition into what they actually are, then that would be great!
Also, if you've seen my questions before, then you'll have likely seen this disclaimer before. Please do not just post full solutions! I want to learn maths and, I believe, the best way to do that is to work out as much of the question as I can myself! A small hint to start with would be appreciated; if I'm still stuck, then I'll ask for a larger hint. Thank you!