Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 6 / b / Solution

Let $d(x,A)$ be the distance to the closed set $A$. Continuity givesEvery variable in the countable infimum is ${\mathcal{F}}_t$-measurable by adaptedness, so the event belongs to ${\mathcal{F}}_t$. Hence $T_A$ is a stopping time.