Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a complete filtration on $(\Omega,\mathcal A,\operatorname P)$ and $A\in\mathcal A$. By the tower property of the conditional expectation, $$M_t:=\operatorname P\left[A\mid\mathcal F_t\right]\;\;\;\text{for }t\ge0$$ is an $\mathcal F$-martingale.
Let $\tau$ be a bounded $\mathcal F$-stopping time. Why are we allowed to apply the optional sampling theorem to conclude $\operatorname P\left[A\mid\mathcal F_\tau\right]=M_\tau$ almost surely?
My problem is that the version of the OST that I know requires $M$ to be right-continuous. So, what am I missing?
On the other hand, $M$ has an $\mathcal F$-optional modificaton $\tilde M$. Is there a version of the OST for optional martingales?