Let $X$ be a strong Markov process on $E$, and $B\in \mathcal B(E)$. Suppose that, for some $x\in E$, $$ P_x(\exists t\ge0 \text{ such that } X_t\in B)=1. $$ My question: Does there exist a stopping time $T$ such that $P_x(X_T\in B)=1$?


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