Hitting open sets Let $(\Omega,\mathscr F,(\mathscr F_t)_{t\geq 0},\mathsf P)$ be a complete filtered probability space and $X = (X_t)_{t\geq 0}$ be a cadlag stochastic process with value in a Polish space $E$. Is it true that for any closed subset $A$ of $E$ the first hitting time
$$
\tau = \inf\{t\geq 0:X_t\in A\}
$$
is a stopping time, i.e. $\{\tau\leq t\}\in \mathscr F_t$ for all $t\geq 0$; and what if $A$ is an open set.
As an example, we can consider $X$ with values in $\mathbb R$. It holds that the first hitting time of a closed half-line 
$$
\tau = \inf\{t\geq 0:X_t\in[K,\infty)\}
$$
is a stopping time. Can we apply the following argument:

Let $\tau = \inf\{t\geq 0:X_t\in (K,\infty)\}$ then
  $$
\{\tau\leq t\} = \bigcup\limits_{n=1}^\infty\{\tau_n\leq t\}\in \mathscr F_t
$$
  where $\tau_n = \inf\{t\geq 0:X_t\in[K+1/n,\infty)\}$ and hence $\{\tau_n\leq t\}\in \mathscr F_t$.

 A: The measurability of hitting times is a subtle and complex problem, 
i.e., a pain in the neck. Of course, the raw $\sigma$-fields are pretty 
much hopeless, even if the process has continuous paths and $A$ is open.
(Take two sample paths that start on the boundary of $A$; one stays put 
while the other immediately heads into $A$. If $\tau_A$ were a stopping time,
 then $1[\tau_A=0]$ should be a function of the initial position only.) 
It is often assumed that the filtration is complete and right continuous,
then things work out better. In this case, for open $A$ and $t>0$ we have 
$$\{\tau_A<t\}=\cup_{r\in Q_t} \{X_r\in A\},$$ 
where $Q_t$ is the set of rational numbers in $[0,t)$. This, and the right-continuity
of the fields shows that $\tau_A$ is a stopping time. 
For general Borel sets, I suggest reading The measurability of hitting times by Richard Bass,
Electron. Comm. Probab. 15 (2010) 99-105; 16 (2011) 189-191 for an elementary (but not easy)
exposition of quite general results about hitting times. Note that even 
Prof. Bass made a subtle error, later corrected.  The paper and correction
 are number 130 on his home page: 
http://bass.math.uconn.edu/biblio.html
