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
$$
\Pr[R(T,A)] = \sup\limits\_{\tau\leq T} \mathbb{E}[I\_A(X\_\tau)]?
$$



Reachability for Markov process
It is right, and not so obvious.


The question of whether or not a Markov process hits
particular sets is usually studied using the concept of capacity.


For a continuous time parameter Markov process taking values in
a general topological state space, this leads to non-trivial problems of measurability.
For instance, for a Borel $A$ there is no guarantee that the set $R(T,A)\in{\cal F}$
where $(\Omega,{\cal F},\Pr)$ is the probability space.
However, under suitable conditions, capacity theory can be used to show that $R(T,A)$ is universally measurable, and hence that $\Pr[R(T,A)]$ makes sense.


Let's assume that the state space and process are "nice";
say, the state space is a locally compact, separable metric space,
and the process has right continuous sample paths.
For fixed $T<\infty$, the formula $\phi(A)=\Pr[R(T,A)]$ defines a Choquet capacity on the
Borel sets $A$. Therefore, $$\phi(A)=\sup(\phi(K): K\subseteq A,\ K\mbox{ compact}).$$


For a compact $K$, define the stopping time $\tau(\omega):=\inf(t\geq 0: X\_t(\omega)\in K)$.
Since the sample paths of $(X\_t)$ are right continuous and $K$ is
closed, we have $R(T,K) = (X\_{\tau\wedge T} \in K).$


Therefore,
$$\Pr[R(T,K)]\leq \mathbb{E}[I\_A(X\_{\tau\wedge T})]\leq \Pr[R(T,A)].$$


Taking the supremum over compact subsets of $A$ gives
$$ \Pr[R(T,A)]=\sup\_{\tau}\ \mathbb{E}[I\_A(X\_{\tau\wedge T})],$$
which gives your desired result. Letting $T\to\infty$ gives the infinite version.


The result hinges on the fact that, as far as the process goes, the Borel set $A$ can be well
approximated from the inside by compact sets.


You can find more details in Chapter I, Section 10 of Blumenthal and Getoor's
 *Markov Processes and Potential Theory*, or in Section 3.3 of Kai Lai Chung's *Lectures from Markov Processes to Brownian Motion*.


