= Adaptedness of a stopped right-continuous process
{title2=$X^T_t=X_{t\wedge T}$}
For an <adapted process> with right-continuous paths and a <stopping time> $T$, $X^T_t=X_{t\wedge T}$ is <adapted>. At fixed $t$, round $t\wedge T$ upward on finite partitions of $[0,t]$. Every approximating value is $\mathcal F_t$-<measurable>, and right continuity gives the limit. Completeness handles a null exceptional set if right continuity holds only <almost surely>. A <martingale> assumption and boundedness of $T$ are unnecessary.
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