Let $[[S]]$ be the graph of the stopping time $S$, and $\pi(A)$ the projection of $A$, which is a subset of $[0,\infty[ \times \Omega$, into $\Omega$.
In the following picture, why would the author want to look at $X_{S\wedge t}$ and not simply at $X_S$? Since $[[S]]\subset A$, when $S< \infty$, we'll have $X$ and $Y$ different.
Also, do we need $P(\pi(A))>0$ just so that we have $P(S<\infty)>0$? What would happen otherwise?
