Let $X = \{ X_t\}$ be a stochastic process on $(\Omega, \mathbb{F}, P)$. Explain briefly why each of the following is or is not a stopping time:
(c) Let $\{A_j \}_{j=1}^n$ be a sequence of measurable sets, and consider the first time $X$ reaches $A_n$, after first reaching $A_{n-1}$, after first reaching $A_{n-2}$, ... , after first reaching $A_1$.
Let $T_i = \inf \{ t \in \mathbb{T}: X_t \in A_i\}$ such that $T_i \le T_{i+1}$ for all $i$, where $\mathbb{T}$ is a set of time. I have to show that $\{T_n \le t\} \in \mathcal{F}_t$ for all $t \in \mathbb{T}$, where $\mathcal{F}_t$ is a filtration of $\mathcal{F}$. My idea is that: $\{T_n \le t\} = \cup_{j = T_{n-1} }^t X^{-1}_j(A_n)$. Then, $X^{-1}_j(A_n) \in \mathcal{F}_t$ for each $j \le t$. But I don't know how to deal with $T_{n-1}$ on the subscript here. Can someone help me?