In Protter's book, they defined a stopping time for the simple process $H_u^k(\omega)$ as $$R_k=\inf\{t:|H_u^k|>\eta\}.$$
Since the simple process is left continuous with right limit and $R_k$ is the hitting time into an open set. I am not really convinced that $R_k$ is indeed a stopping time. How can I see that? Any help is appreciated
Using the definition of the process and the left-continuity of $H$ $$\{R_k > t\} = \bigcup_{s \le t,~ s\in\mathbb Q} \left\{H_s^k \le \eta\right\}$$ Since $\mathbb Q$ is countable and $\left\{H_s^k \le \eta\right\} \in \mathcal {F}_t, \forall s \le t$, we have $\{R_k > t\} \in \mathcal F_t$ which proves that $R_k$ is a stopping time.