We let $\displaystyle \sigma$ and $\displaystyle \tau$ be 2 stopping times, where $\displaystyle \sigma\leq\tau$.

For a set $\displaystyle A\in F_\sigma$, if we define

$\displaystyle \sigma'(\omega)=$
$\displaystyle \sigma(\omega)$ if $\displaystyle \omega\in{A}$,
$\displaystyle \tau(\omega)$ if $\displaystyle \omega\in{A^c}$.




Then $\displaystyle \sigma'(\omega)$ is still a stopping time. Can anyone explain the reason? Thanks!