A stochastic analysis problem about martingale I am studying Le Gall's "Brownian Motion,Martingales, and Stochastic Calculus"and come across the following sentence:
In the definition of a continuous local martingale starting from 0, one can
replace “uniformly integrable martingale” by “martingale” (indeed, one can
then observe that $M^{Tn\wedge n}$is uniformly integrable, and we still have $T_n \wedge n \uparrow\infty$.)
How to understand that "one can
replace “uniformly integrable martingale” by “martingale” "I think they are not equivalent?I am confused about that.

If $M$ is a martingale, then the deterministic stopping time $\omega \mapsto n$ is a stopping time for the process. Furthermore, $M^n$ is a uniformly integrable martingale.
If now by the first definition there exists stopping times $\tau_n$ such that $({M^n})^{\tau_n} = M^{\tau_n\land n}$ is an uniformly martingale for each $n$, then this also holds true for the (not necessarily uniformly integrable) martingale $M$, since $n\land \tau_n$ is again a stopping time.
