by the definition of localization of martingale, one require for a sequence of stopping time $\tau_n\to \infty$ the $$1_{\tau_n>0}X_{\min\{t,\tau_n\}}$$ to be a martingale.
My question is:
- Why it is necessary to multiply with the indicator function $1_{\tau_n>0}$ in the definition.
- Why do I require $\tau_n\to \infty$ for $n\to \infty$?