Given the following proof of the compensated Poisson process being a Martingale
Why does the proof start with $E[N(t)-\lambda t|N(s)]$ when the question asks to prove that $X(t)$ is a Martingale? Shouldn't it start with $E[N(t)-\lambda t|N(s)-\lambda s]$?
I see that the solution shows that when you plug in the 2nd line of the solution into the left side of $E[N(s)|N(t)]-\lambda t$ you will get $N(s)-\lambda s$ which is equivalent to $X(s)$ but I'm still confused as to why the proof is starting with, essentially, $E[X(t)|X(s)+\lambda s]$ to prove the Martingale when the given definition seems to imply you should start with $E[X(t)|X(s)]$ and then try to get to a final answer of $X(s)$
Conditioning on $X(s)+\lambda s$ is the same as conditioning on $X(s)$ since $\lambda s$ is not random. I do not know your background on probability theory, but in an elementary explanation, the information $X(s)+\lambda s$ and $X(s)$ carry is exactly the same since the two differ by a deterministic quantity, so knowing one can recover another. In measure-theoretic probability terms. the sigma-fields generated by the two are exactly the same.