In order to generalize predictable processes, one defines the predictable sigma algebra: $$ \mathbb{P} = \sigma(\{(s,t] \times A \mid A \in F_s ∧ s < t\} ∪ \{\{⊥\} \times A \mid A \in F_⊥\}) $$
If the index set is the natural numbers, $\mathbb{P}$-measurability should be equivalent to saying that $X_n$ is $F_{n-1}$ measurable for all $n \ge 1$. I am trying to show this fact but I am stuck at the following step:
Assuming $X$ is $\mathbb{P}$-measurable, we have for any borel set $S$: $$ X^{-1} (S) \cap ((n,n+1] \times\Omega) \in \mathbb{P} $$ hence, $$ (n,n+1] \times (X_{n + 1}^{-1}(S)) \in \mathbb{P} $$ From this, I want to show:
$$X_{n + 1}^{-1}(S) \in F_n$$
I tried to show this via induction, on the inductively defined sigma algebra $\mathbb{P}$, however I am stuck at the complement step. I don't know what is the easiest way to show this. Any help is appreciated!