Shift of Stopping Time is Stopping Time

Theorem

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

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

Let $t$ be a extended natural number.

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

Proof

By Constant Function is Stopping Time, $t$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

By Sum of Stopping Times is Stopping Time, $T + t$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

$\blacksquare$