Problem 2.22 in Karatzas' & Shreve's Brownian Motion and Stochastic Calculus asks to
Prove that if $S$ is an optional time and $T$ is a positive stopping time with $S \le T$ and $S < T$ on $\{S < \infty\}$, then $\mathcal{F}_{S+} \subset \mathcal{F}_T$.
My question is: why is "positive" emphasized in the problem statement, what fails without this assumption? For reference, I'd solve this as follows:
Let $A ∈ \mathcal F_{S+}$. Then for any $t \geq 0$, $A ∩ \{ S < t \} ∈ \mathcal F_t$ by problem 2.21, hence $$A ∩ \{ T \leq t \} = A ∩ \{ T \leq t \} ∩ \{ S < ∞ \} = \underbrace{A ∩ \{ S < t \}}_{∈ \mathcal F_t}∩ \underbrace{\{T \le t\}}_{ ∈ \mathcal F_t } ∈ \mathcal F_t, $$ where the first equality uses $S \leq T$, so that $\{ T \le t \} ⊆ \{ S < ∞ \}$, and the second equality uses that $\{ S < ∞ \} ⊆ \{ S < T \}$, so that $\{ T \leq t \} ∩ \{ S < ∞ \} = \{ T \leq t \} ∩ \{ S < t \}$. Since $t \ge 0$ was arbitrary, $A ∈ \mathcal F_T$ follows.