Closed-level hitting times of continuous adapted processes ($T=\inf \{s:X_s\ge \lambda \}$)

For a continuous adapted process $X$ and $T=\inf \{s\ge 0:X_s\ge \lambda \}$,Continuity makes the supremum on $[0,t]$ attained and equals its supremum on this countable dense set. Each displayed event belongs to ${\mathcal{F}}_t$, so $T$ is a stopping time without requiring right-continuity of the filtration.

// Target: martingale.bigb