Do not assume usual hypothesis. Let $A$ be a set measurable with respect to optional sigma-algebra. Then is debut of $A$, $D_A$, a stopping time with respect to $F$ or $F_+$ filtration?
I know that if filtration is complete then the debut of a progressive set is a $(F)$ stopping time. But in case of not necesseraliy complete filtration, by assuming stronger condition of optionality, I feel that may give the same result.