Measurability of Last Exit Time of a Discrete Time Stochastic Process

Question

Suppose we have a discrete-time stochastic process $\{X_t\}$ defined on a space $(\Omega,\mathcal{F})$ equipped with the probability measure $\mathbb{P}$. Suppose we know that $X_t \rightarrow 0$ almost surely. Let $T(\epsilon)$ be defined as $T(\epsilon)=sup\{t\in\mathbb{N}: |X_t| \geq \epsilon\}$. My question is whether $T(\epsilon)$ is measurable w.r.t. the $\sigma$-algebra $\mathcal{F}$. This appears to be true, but a proof or a reference to a proof would be very helpful.

Answer 1

To verify a positive integer valued random variable $N$ is measurable, it suffices to show the sets $\{N=n\}$ are measurable, for $n\in \mathbb N$.

$$ \{T(\epsilon)=n\}=\{|X_n|\ge \epsilon\}\cap \bigcap_{t=n+1}^{\infty} \{|X_t|< \epsilon\} $$ The RHS is measurable because it is a countable intersection of measurable events.