Let's first state the theorem
$\forall M$ continuous local martingale, there exists a unique increasing continuous process $\langle M\rangle $ zero at $t=0$ and such that $M^2-\langle M \rangle $ is again a continuous local martingale. Further for all stopping times $\tau$, we have $\langle M^\tau\rangle = \langle M \rangle^\tau$.
The last statement is not clear for me. I know that $L:=M^2-\langle M \rangle $ is a local martingale. Then we look at $L^\tau=(M^\tau)^2-\langle M \rangle^\tau $, and we apply the stopping theorem. Why do we can apply the stopping theorem? For that, $L$ should be uniformly integrable, or $\tau$ must be finite?
thanks for your help