Pointwise Supremum of Stopping Times is Stopping Time

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {T_n}_{n \in \N}$ be a sequence of stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$.

Let:


 * $\ds T = \sup_{n \in \N} T_n$

be the pointwise supremum of the $\sequence {T_n}_{n \in \N}$.

Then $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Proof
We have, for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$:


 * $\map T \omega \le t$ $\map {T_n} \omega \le t$ for all $n \in \N$.

That is:


 * $\ds \set {\omega \in \Omega : \map T \omega \le t} = \bigcap_{n \in \N} \set {\omega \in \Omega : \map {T_n} \omega \le t}$

for each $t \in \Z_{\ge 0}$.

Now fix $t \in \Z_{\ge 0}$.

Since each $T_n$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$, we have:


 * $\set {\omega \in \Omega : \map {T_n} \omega \le t} \in \FF_t$

for each $n \in \N$.

Then, since $\FF_t$ is a $\sigma$-algebra and $\sigma$-algebras are closed under countable intersection, we have:


 * $\ds \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$

So $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.