Can someone check if the proof of theorem 8.13 of the book Martingales and Stochastic Analysis by James Yeh is correct (link here: https://goo.gl/ivxJnv, the Google Book version).
Note line 11, page 135 - "On the set $\{S \leq t \}$, we have $X_{S\wedge t} = X_t$ ...". Isn't that clearly wrong? I think the author confused $\{S\leq t\}$ with $\{S> t\}$ and the argument is wrong from line 3. Line 7 also claims $A \cap \{S > t \} \in \mathcal{F}_t$ since $A \in \mathcal{F}_S$, isn't this also wrong? $A \in \mathcal{F}_S$ implies $A \cap \{S\leq t\} \in \mathcal{F}_t$ but not necessarily $A \cap \{S>t\}$ as well right? I think the argument can be fixed by replacing $\{S > t\}$ with $\{S\leq t\}$ and vice versa (with the appropriate change in the argument that follows the switch). Can anyone confirm please?