On events connected with reaching a set by sample paths of a stochastic process

Let $\Gamma$ be a subset in the state space of a stochastic process $x_t$. Let $I$ be an interval on the real line $T$, and $D(I)$ be the event $\{x_t\in\Gamma\ \text{at some}\ t\in I\}$. Such a system of events $D(I)$ satisfies conditions 1.A–1.B.
Under some assumptions, in the Markov case, all such systems are described. The main result is applied to the analysis of a special $\sigma$-field in the space $T\times\Omega$.