consequence of the definition of *uniqueness*.

\[uni prop\] Let $\Lambda$ be a set of ${\mathfrak{F}}$-stopping times such that ${[\hspace{-0.06cm}[}0, \rho{]\hspace{-0.06cm}]}\subseteq A^*\in \mathcal{I}({\mathfrak{F}})$ for all $\rho \in \Lambda$, let $A\in \mathcal{I}({\mathfrak{F}})$ such that $A^* \subseteq A$, and let $(B(h),C,\nu)$, $(B^*(h), C^*, \nu^*)$ be two candidate triplets which coincide on $A^*$. Then the following is equivalent:

1.  The SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$ satisfies $\Lambda$-uniqueness.

2.  The SMP associated with $(\mathbb{H}; A^*; \eta; X; B^*(h), C^*, \nu^*)$ satisfies $\Lambda$-uniqueness.

Main Results {#section: Martingality of local martingales}
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Let $(\Omega, \mathcal{F}, {\mathfrak{F}}, {{\mathbf{P}}})$ be a given filtered probability space with right-continuous filtration ${\mathfrak{F}}= (\mathcal{F}_t)_{t \geq 0}$. We are interested in the martingality of an $\mathbb{R}^+$-valued local $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale $Z$ with ${{\mathbf{P}}}$-a.s. $Z_0 = 1$. Later we will assume that $Z$ is driven by a Hilbert-space-valued semimartingale. We describe the martingality of $Z$ by convergence properties of sequences of stopping times satisfying the following convention which we impose from now on.

\[conv\] Let $(\rho_n)_{n \in \mathbb{N}}$ be an increasing sequence of ${\mathfrak{F}}$-stopping times such that $Z^{\rho_n}$ is a uniformly integrable $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale for all $n \in \mathbb{N}$.

A sequence as in Convention \[conv\] always exists, namely any localizing sequence 