of $Z$. It is important to note that the sequence in Convention \[conv\] need not to increase ${{\mathbf{P}}}$-a.s. to infinity. In Appendix \[An Integrability Condition to Indentify\] we present an integrability condition to identify such sequences, c.f. [@CFY] for a related approach based on a Novikov-type condition. In Section \[Martingality in Terms of SMPs\], we relate martingality to existence and uniqueness properties of SMPs. In Section \[Martingality on Standard Systems\], under topological assumptions on the underlying filtered space, we derive sufficient conditions for the martingality of $Z$ in terms of an almost sure convergence of  $(\rho_n)_{n \in \mathbb{N}}$.

Martingality in Terms of SMPs {#Martingality in Terms of SMPs}
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We consider five situations: Firstly, we derive a condition for the strict local martingality of $Z$ in terms of existence properties of SMPs. Secondly, we impose the assumption of $\Lambda$-uniqueness for a modified SMP. This additional structure enables us to investigate the martingality of $Z$ by examining the limit behavior of the sequence of stopping times $(\rho_n)_{n \in \mathbb{N}}$ under a solution to an SMP. Thirdly, we construct an explicit sequence $(\rho_n)_{n \in \mathbb{N}}$ driven solely by $Z$. This results in martingality conditions only depending on the path of $Z$. Fourthly, 