What is the definition of a local martingale on $[0,T]?$
I guess the definition should be: $M = (M_t)_{t \in [0,T]}$ is a local martingale if there exists a localizing sequence $\tau_n$ such that for all $n$ the process $M^{\tau_n}$ is a martingale.
A sequence of stopping times $(\tau_n)_n$ is a localizing sequence if $P$-almost surely $\tau_n \leq \tau_{n+1}$ and $\lim_{n \to \infty} \tau_n = T.$
Is this definition correct?