My question concerns the book Lectures from Markov Processes to Brownian Motion by Kai Lai Chung, more precisely the remark at the bottom of page 76:
We prove later in paragraph 3.3 that on $\{ t < \zeta \}$ we have $X_{t-} \neq \partial$, namely $X_{t-} \in \mathbf{E}$. For a Feller process this is implied by theorem 7 of paragraph 2.2.
To put things in context, the book studies Markov processes with values in a metric space $\mathbf{E}$, to which one adjoins a point at infinity $\partial$, yielding the one-point (Alexandrov) compactification $\mathbf{E}_{\partial}$. $\zeta$ is defined in the statement of theorem 7, paragraph 2.2:
$$\zeta(\omega) \equiv \inf\left\{ t \geq 0 \,;\, X_{t-}(\omega) = \partial\textrm{ or }X_t(\omega) = \partial \right\}$$
The statement of the aforementioned theorem 7 is as follows:
Let $\{X_t, \mathcal{F}_t\}$ be a Feller process with right-continuous paths having left limits. Then we have almost surely $X(\zeta + t) = \partial$ for all $t \geq 0$, on the set $\{ \zeta < \infty \}$.
I am not sure I understand why we need to invoke theorem 7 to prove the statement in the remark. Indeed, if $X_{t-} = \partial$ for some $t < \zeta$, doesn't it simply contradict the definition of $\zeta$? Could you shed some light on this point?
Checking another reference on the general theory of Hunt processes: Fukushima, Dirichlet Forms and Markov Processes the life-time is defined there as
$$\zeta(\omega) \equiv \inf\{ t \geq 0 \,;\, X_t(\omega) = \partial \}$$
It seems the remark now makes sense if the authors thought about this definition rather than the one given in theorem 7. I therefore assume the problem comes from an inconsistency in the definitions/notations within the book.