\infty} {{\mathbf{E}}}\big[{{\mathbf{E}}}\big[Z_{\rho_n} {\mathbf{1}}_{\{\rho_n > t\}}|\mathcal{F}_{t \wedge \rho_n}\big]\big]
= \lim_{n \to \infty} {{\mathbf{Q}}}_n(\rho_n> t).\end{aligned}$$ This finishes the proof.

If we relinquish on an equivalence statement we can relax the assumption in Lemma \[ruf mimic\] that $(\rho_n)_{n \in \mathbb{N}}$ is a localizing sequence.

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

1.  If holds for all $t \geq 0$, then $Z$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale.

2.  If $$\begin{aligned}
    \lim_{n \to \infty} {{\mathbf{Q}}}_n(\rho_n = \infty) = 1,\textit{ where } {{\mathbf{Q}}}_n = Z_{\rho_n} \cdot {{\mathbf{P}}},\end{aligned}$$ then $Z$ is a uniformly integrable $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale.

We first prove (i) and obtain that $$\begin{aligned}
1 \geq {{\mathbf{E}}}[Z_t] \geq {{\mathbf{E}}}[Z_t \lim_{n \to \infty} {\mathbf{1}}_{\{\rho_n> t\}}] = \lim_{n \to \infty} {{\mathbf{E}}}[Z_t {\mathbf{1}}_{\{\rho_n > t\}}] = \lim_{n \to \infty} {{\mathbf{Q}}}_n(\rho_n > t) = 1,\end{aligned}$$ by the same argumentation as in the proof of Lemma \[ruf mimic\]. Therefore, as $Z$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-supermartingale, $Z$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale.

We now turn to (ii). $Z$ is a uniformly integrable $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale if ${{\mathbf{E}}}[Z_\infty] \geq 1$, where $Z_\infty := \lim_{t \to \infty} Z_t$ exists due to the supermartingale convergences theorem. 