Let $X$ be a cadlag stochastic process. If $X$ is continuous, then I already know that
$\inf\{t\geq 0: X_t \in C\}$ is a stopping time whenever $C$ is closed in $\mathbb{R}$. What if $X$ is only cadlag? (For every $\omega$)?
Furthermore, in the notes here: http://www.statslab.cam.ac.uk/~beresty/teach/StoCal/sc3.pdf I am having trouble surrounding defn 3.8 and the line before it on p. 39. For the line before it, here is a seemingly legal cadlag adapted process which is necessarily previsible, but not locally bounded. I take my favorite unbounded RV, and the process is constantly equal to that RV. Same objection to the idea in defn 3.8 that $M^{{S'}_n}$ is $L^2$ bounded. (I am mentally removing all the a.s.. Some of them are meaningful but seemingly unnecessary, but some of them are in the presence of no probability measure, such as the second one in defn 3.7)
Possibly being obscured by the problems before this point, I am having trouble making sense of equation (3.44). The result of the integral is technically an equivalence class of indistinguishable processes. So is he defining the whole process at once up to indistinguishability, and then recovering the slices for a fixed $t$ by evaluating into $L^2$ equivalence classes?
An answer to any subset of these is appreciated.