Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / A / 1 / b / ii

For $A\in {\mathcal{F}}_S$, writeThe first two factors together belong to ${\mathcal{F}}_k$, since $A\cap \{S=k\}$ is the difference of its intersections with $\{S\le k\}$ and $\{S\le k-1\}$. Also $\{k\le T\le n\}\in {\mathcal{F}}_n$ by the stopping time property. The filtration inclusion puts every term, and therefore the union, in ${\mathcal{F}}_n$. By definition,