 Pasting ordered stopping times

If $S\le T$ are stopping times and $A\in {\mathcal{F}}_S$, then $U=S{1}_A+T{1}_{A^c}$ is a stopping time. The key identity isThis uses the ordering $S\le T$: without it, $A$ need not be known when $T$ occurs. Such pastings test the martingale identity through expectations at bounded stopping times.