Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra

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

Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

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

Let $X_T$ be the adapted stochastic process $\sequence {X_n}_{n \ge 0}$ stopped at $T$.

Let $\FF_T$ be the stopped $\sigma$-algebra associated with $T$.

Then $X_T$ is $\FF_T$-measurable.

Proof
We have that if $\map T \omega = t$ for $\omega \in \Omega$ and $t \in \Z_{\ge 0}$ then:


 * $\map {X_T} \omega = \map {X_t} \omega$

We aim to show that for each Borel set $A \subseteq \R$:


 * $\ds \set {\omega \in \Omega : \map {X_T} \omega \in A} \in \FF_T$

That is, we want to show that:


 * $\set {\omega \in \Omega : \map {X_T} \omega \in A} \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$

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

Let $t \in \Z_{\ge 0}$.

We have that:


 * $\ds \set {\omega \in \Omega : \map {X_T} \omega \in A} \cap \set {\omega \in \Omega : \map T \omega \le t} = \bigcup_{s \in \Z_{\ge 0}, \, 0 \le s \le t} \set {\omega \in \Omega : \map T \omega = s} \cap \set {\omega \in \Omega : \map {X_s} \omega \in A}$

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


 * $\set {\omega \in \Omega : \map T \omega = s} \in \FF_s$

Since $\sequence {\FF_n}_{n \ge 0}$ is a filtration of $\sigma$-algebra and $s \le t$:


 * $\set {\omega \in \Omega : \map T \omega = s} \in \FF_t$

Since $\sequence {X_n}_{n \ge 0}$ is $\sequence {\FF_n}_{n \ge 0}$-adapted, we have that $X_s$ is $\FF_s$-measurable.

So, we have:


 * $\set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_s$

and so:


 * $\set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_t$

Since $\FF_t$ is closed under finite intersection, we have:


 * $\set {\omega \in \Omega : \map T \omega = s} \cap \set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_t$

Since $\FF_t$ is closed under finite union, we have:


 * $\ds \bigcup_{s \in \Z_{\ge 0}, \, 0 \le s \le t} \set {\omega \in \Omega : \map T \omega = s} \cap \set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_t$

and so:


 * $\set {\omega \in \Omega : \map {X_T} \omega \in A} \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$

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

We therefore have:


 * $\set {\omega \in \Omega : \map {X_T} \omega \in A} \in \FF_T$

So $X_T$ is $\FF_T$-measurable.