Hitting time definition clarification

Question

Defintion of hitting time in my textbook is written as so:

Let $(X_n)_{n\geq 0}$ be a Markov $(\lambda, P)$. The hitting time of a subset $A \subseteq I$ is a stopping time denoted by $H^A$ such that:

$$H^A(\omega)= inf\{n\geq 0 : X_n(\omega) \in A \}$$

This is basically "the first time you hit A".

Now the notation then that the book provides me with is:

$$h^A_i = \Bbb P_i(H^A \lt \infty)$$ and it mean that this is the probability that of hitting $A$ given I started at state $i$. The only thing I don't understand here is the "$H^A \lt \infty$" part. What does this intuitively mean?

Answer 1

The set $\{ H^A < \infty \} $ is $\{ \omega, H^A(\omega) < \infty \} = \{\omega, \inf(n, X_n(w) \in A) < \infty\}$ where it is assumed that $\inf ( \emptyset) = \infty$.

Thus $\{H^A < \infty \}$ is the set of all the possible outcomes (the $\omega$) for which there is at least of $n \in \mathbb N$ such that $X_n(\omega) \in A$