Let $(X_n)_n$ be a sequence of random variables and $τ$ be an a.s.-finite (i.e. $\Bbb P[τ <∞] = 1$) stopping time w.r.t. $(F_n)$. Prove that $X_τ(w):=X_{\tau(w)}(w)$, for any $w$ such that $\tau(w)<\infty$, is a random variable.
I basically say that for $A \in \mathcal B(\Bbb R)$, we have that $X_\tau^{-1}(A)=\bigcup_{n=1}^\infty \{ \tau^{-1}(n) \cap X_n^{-1}(A) \}$ which is a countable union of measurable sets so $X_\tau$ is a measurable function i.e. a random variable.
I did not use the fact that $\Bbb P[τ <∞] = 1$. Even if there was a set $S \subset \Bbb N$ with $\Bbb P(S)>0$ and $\tau(w)=\infty \forall w \in S (\implies P[τ =∞] > 0)$ my reasoning is still valid I guess so I don't see why would one needs the assumption $P[τ <∞] = 1$