The factor \(\mathbb{1}_{\left\{\sigma_{n}>0ight\}}\) is used in the definition of a local martingale, in order to avoid integrability requirements for \(M_{0}\). More precisely, if \(\left(\sigma_{n}ight)_{n \geqslant 1}\) is a localizing sequence, then we have

\(M_{0} \in L^{1}(\mathbb{P}) \&\left(M_{t}ight)_{t \geqslant 0} \quad\) is a local martingale \(\Longleftrightarrow\left(M_{\sigma_{n} \wedge t}ight)_{t \geqslant 0} \quad\) is a martingale.