In my book, it states that any constant, including, $\infty$, is a stopping time. But why is this? I thought the reason might be that a constant is measurable with regards to the trivial sigma-algebra, i.e. $\{\emptyset, \Omega\}$ and since $\emptyset \in \mathcal{F}_n$ the results follow? Or what is the reason?
Thanks.
Let $\tau=k,\,k \in \mathbb{N}\cup \{\infty\}$. Recalling the definition, $\tau$ is a $\mathscr{F}_n$-stopping time if $\{\tau\leq n\}\in \mathscr{F}_n,\,\forall n \in \mathbb{N}$. Now if $k<\infty$ we have $$\{\tau\leq n\}=\begin{cases}\Omega&n\geq k\\ \emptyset &n<k \end{cases}\in \mathscr{F}_n,\,\forall n \in \mathbb{N}$$ while if $k=\infty$, $\{\tau\leq n\}=\emptyset$ for all $n$ so $\tau $ is always a stopping time.