is a positive $\mathcal{F}_0$-measurable and ${{\mathbf{P}}}$-integrable random variable by replacing $Z$ with $Z/Z_0$. The following lemma is in the spirit of Theorem 1.3.5 in [@SV], Lemma III.3.3 in [@JS], and Corollary 2.1 in [@BR16].

\[ruf mimic\] The following is equivalent:

1.  $Z$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale.

2.  There exists an $({\mathfrak{F}}, {{\mathbf{P}}})$-localization sequence $(\rho_n)_{n \in \mathbb{N}}$ of $Z$ such that $$\begin{aligned}
    \label{difficult to check}
    \lim_{n \to \infty} {{\mathbf{Q}}}_n (\rho_n> t) = 1\ \textit{ for all } t \geq 0,\end{aligned}$$ where ${{\mathbf{Q}}}_n := Z_{\rho_n} \cdot {{\mathbf{P}}}$, i.e. ${{\mathbf{Q}}}_n({\operatorname{d}\hspace{-0.003cm}}\omega) = Z_{\rho_n}(\omega){{\mathbf{P}}}({\operatorname{d}\hspace{-0.003cm}}\omega)$, for $n \in \mathbb{N}$.

3.  For all $({\mathfrak{F}}, {{\mathbf{P}}})$-localizing sequences $(\rho_n)_{n \in \mathbb{N}}$ of $Z$ the convergence holds.

The implication (iii) $\Longrightarrow$ (ii) is trivial. We show that (ii) $\Longrightarrow$ (i) $\Longrightarrow$ (iii). $Z$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale if and only if for all $t \geq 0$ we have ${{\mathbf{E}}}[Z_t] = {{\mathbf{E}}}[Z_0] = 1$. Let $(\rho_n)_{n \in \mathbb{N}}$ be an arbitrary $({\mathfrak{F}}, {{\mathbf{P}}})$-localizing sequence. Due to monotone convergence, Doob’s stopping theorem and $\{\rho_n > t\} \in \mathcal{F}_{t \wedge \rho_n}$, we have $$\begin{aligned}
{{\mathbf{E}}}[Z_t] &= {{\mathbf{E}}}\big[Z_t \lim_{n \to \infty} {\mathbf{1}}_{\{\rho_n > t\}}\big] = \lim_{n \to \infty} {{\mathbf{E}}}\big[Z_{t \wedge \rho_n} {\mathbf{1}}_{\{\rho_n> t\}}\big] = \lim_{n \to \infty} {{\mathbf{E}}}\big[{{\mathbf{E}}}\big[Z_{\rho_n}|\mathcal{F}_{t \wedge \rho_n}\big]{\mathbf{1}}_{\{\rho_n> t\}}\big]
\\&= \lim_{n \to 