I am trying to understand a detail of the proof of Theorem VI.82 of the book by Dellacherie and Meyer.
Theorem Let $M$ be a local martingale and let $N_t=\sup_{s\leq t}\left|M_s\right|$. Then $N$ is locally integrable.
The proof is elegant and (looks) simple. The authors consider the localizing sequence $R_n$ that makes the stopped process $M^{(R_n)}$ an integrable martingale and the sequence $P_n$ given by $P_n=\inf\{t\geq0\mid \left|M_t\right|\geq n\}$. Then they define $$ S_n = R_n\wedge P_n $$ We clearly have that $|M|<n$ on the stochastic interval $[0,S_n[$. This should imply that $$ N_{S_n}\leq n $$ and, for me, it would be enough to conclude that $N$ is locally integrable (because it is also increasing, so once stopped at $S_n$ it is bounded). Nevertheless, they say that $$ N_{S_n}\leq n \vee |M_{S_n}|\quad (1) $$ having proved before (it is not difficult) that $|M_{S_n}|$ is integrable. Any idea of how to obtain the $(1)$ and why we really need it?
ps = it looks like we do not really need the property of being a martingale for the stopped process, but just integrable.
Well, $M$ may (and often will) have a jump at $S_n$, so it is quite possible that $|M_{S_n}|>n$. So actually $$ N_{S_n} \le |M_{S_n}| \vee \sup_{s<S_n} |M_s| \le |M_{S_n}| \vee n. $$