Let $X$ be a Markov process (in continuous or discrete time) and define an event $$ R(T,A) = (\exists t\leq T: X_t \in A). $$ I have seen in one paper that $$ \Pr[R(\infty,A)] = \sup\limits_{\tau} \mathbb{E}[I_A(X_\tau)], $$ where $I_A(x) = 1$ for $x\in A$ and $I_A(x)=0$ otherwise is an indicator function and supremum is taken over all stopping times $\tau:\Pr[\tau<\infty] = 1$.

Unfortunately the auothor did not provide a proof it, so I wonder is it right (and so obvious)? Also, does it imply that $$ \Pr[R(T,A)] = \sup\limits_{\tau\leq T} \mathbb{E}[I_A(X_\tau)]? $$

Let $X$ be a Markov process (in continuous or discrete time) and define an event $$ R(T,A) = (\exists t\leq T: X_t \in A). $$ I have seen in one paper that $$ \Pr[R(\infty,A)] = \sup\limits_{\tau} \mathbb{E}[I_A(X_\tau)], $$ where $I_A(x) = 1$ for $x\in A$ and $I_A(x)=0$ otherwise is an indicator function and supremum is taken over all stopping times $\tau:\Pr[\tau<\infty] = 1$.

Unfortunately the author did not provide a proof it, so I wonder is it right (and so obvious)? Also, does it imply that $$ \Pr[R(T,A)] = \sup\limits_{\tau\leq T} \mathbb{E}[I_A(X_\tau)]? $$