= \bigcup_{n \in \mathbb{N}} {[\hspace{-0.06cm}[}0, \tau_n{]\hspace{-0.06cm}]}.\end{aligned}$$

Sets of interval type are ${\mathfrak{F}}$-predictable. With a little abuse of terminology, we call the sequence $(\tau_n)_{n \in \mathbb{N}}$ *announcing sequence for $A$*. The set of all random sets of interval type is denoted by $\mathcal{I}({\mathfrak{F}})$. Let $\mathcal{C}$ be a class of $E$-valued processes. For $A \in \mathcal{I}({\mathfrak{F}})$ we set $$\begin{aligned}
\mathcal{C}^{A} &:= \big\{ X : A \to\ E : \exists \textup{ announcing sequence } (\tau_n)_{n \in \mathbb{N}} \textup{ for } A \textup{ s.th. } 
X^{\tau_n} \in \mathcal{C}\ \forall n \in \mathbb{N}\big\}.\end{aligned}$$ Since $(\omega, t \wedge \tau_n (\omega)) \in A$ for all $(\omega, t) \in {[\hspace{-0.06cm}[}0, \infty{[\hspace{-0.06cm}[}$, we have $X^{\tau_n} : {[\hspace{-0.06cm}[}0, \infty{[\hspace{-0.06cm}[}\ \to E$ for all $X \in \mathcal{C}^A$.

\[subinterval semimartingale\] Let $\mathcal{C}$ be stable under stopping and localization, $X \in \mathcal{C}^{A}$, and $\rho$ an ${\mathfrak{F}}$-stopping time such that ${[\hspace{-0.06cm}[}0, \rho{]\hspace{-0.06cm}]}\subseteq A$. Then $X^{\rho}\in \mathcal{C}$.

The claim follows identically to [@HWY], Theorem 8.20.

Examples of classes which are stable under stopping and localization are the class of $E$-valued local $({\mathfrak{F}}, {{\mathbf{P}}})$-martingales, denoted by $\mathcal{M}_\textup{loc}(E, {\mathfrak{F}}, {{\mathbf{P}}})$, and the class of $E$-valued $({\mathfrak{F}}, {{\mathbf{P}}})$-semimartingales, denoted by $\mathcal{S}(E, {\mathfrak{F}}, {{\mathbf{P}}})$, where $E$ is a real separable Hilbert space. More generally, if $\mathcal{C}$ is stable under 