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

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

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

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