I'm studying Stochast integration and stochastic differential equation by Protter and got confused by the solution of the very first exercise provided here.
So
Exercise 1. Let $S,T$ be stopping times, $S \leq T$ a.s. Show $\mathcal{F}_S \subset \mathcal{F}_T$
The provided solution starts like this:
$\forall A \subset \mathcal{F}_S, \forall t \geq 0, A \cap \{T \leq t\} = (A \cap \{S \leq t\}) \cap \{T \leq t\}$ since $\{T \leq t\} \subset \{S \leq t\}$
But $S \leq T$ almost surely. It means that there is a set $N \subset \mathcal{F}_0$ (usual hypotheses) with $\mathbf{P}(N) = 0$ such that $ \forall \omega \in N : S(\omega) > T(\omega)$ so $\exists t_0 : \{T \leq t_0\} \not\subset \{S \leq t_0\}$
Thus the start of the proof need to be corrected like that:
$\forall A \subset \mathcal{F}_S, \forall t \geq 0, A \cap \{T \leq t\} = (A \cap (\{S \leq t\} \cup N)) \cap \{T \leq t\}$ since $\{T \leq t\} \subset \{S \leq t\} \cup N$
The rest of the solution works if we replace $\{S \leq t\}$ by $\{S \leq t\} \cup N$:
Since $A \cap (\{S \leq t\} \cup N) \subset \mathcal{F}_t$ (since $N \subset \mathcal{F}_0 \subset \mathcal{F}_t$) and $\{T \leq t\} \subset \mathcal{F}_t$ we have $A \cap \{T \leq t\} \subset \mathcal{F}_t$ so $\mathcal{F}_S \subset \mathcal{F}_T$
Am I right or it is an unnecessary complication of the correct solution?