stopping, then its localized class $\mathcal{C}_\textup{loc}$ is stable under stopping and localization, c.f. [@JS], Lemma I.1.35. We obtain two obvious consequences of Proposition \[subinterval semimartingale\]:

\[ordinary semimartingale\] Let $\mathcal{C}$ be stable under stopping and localization.

1.  $Y\in\mathcal{C}^{A}$ if and only if for all ${\mathfrak{F}}$-stopping times $\rho$ such that ${[\hspace{-0.06cm}[}0, \rho{]\hspace{-0.06cm}]}\subseteq A$, $Y^{\rho}\in \mathcal{C}$.

2.  $\mathcal{C}^{{[\hspace{-0.06cm}[}0, \infty{[\hspace{-0.06cm}[}} = \mathcal{C}$.

To construct non-negative local martingales from semimartingale on sets of interval type the following extension due to [@J79] proves itself useful in the Sections \[MSE\] and \[Martingality of Generalized Stochastic Exponentials\]. For $X\in \mathcal{C}^A$, where $\mathcal{C}$ is a class of $[- \infty, \infty]$-valued processes, we define the extension $\widetilde{X}$ by $$\begin{aligned}
\label{extension}
\widetilde{X}_t := \begin{cases} 
X_t&\textup{ on } \{t < \tau\},\\
X_{\tau},&\textup{ on } \{t \geq \tau\} \cap G^c,\\
\liminf_{s \uparrow \tau, s \in A \cap \mathbb{Q} \times \Omega} X_s,&\textup{ on } \{t \geq \tau\} \cap G,
\end{cases}\end{aligned}$$ where $
\tau := \lim_{n \to \infty} \tau_n$ and $G := \bigcap_{n \in \mathbb{N}} \{\tau_n < \tau < \infty\}
$ for an arbitrary announcing sequence $(\tau_n)_{n \in \mathbb{N}}$ for  $A$.

If ${[\hspace{-0.06cm}[}0, \rho{]\hspace{-0.06cm}]}\subseteq A$ for an ${\mathfrak{F}}$-stopping time $\rho$, then $\widetilde{X}^\rho = X^{\rho}$.

Clearly, in general $\widetilde{X} \not \in \mathcal{C}$, however, if $X \in 