I'm reading about Markov chain and stopping times from Varadhan's book on Probabality (See also his notes on his website).
Roughly speaking, the data for Markov Chain is a state space with a $\sigma$-algebra $(\mathcal X,\mathcal{F})$, an initial probability $\mu_0$ on $\mathcal X$, and transition probabilities $\{\pi_n\}_{n \in \mathbb{N}}$ on $\mathcal X$ which depend measurably on previous states.
From this one can construct a space of Markov chains with a probability $(\Omega:=\mathcal X^{\{0\}\cup \mathbb{N}},\mathcal{F}^\infty:=\mathcal{F}^{\{0\}\cup \mathbb{N}}, P)$.
Denote the $\sigma$-algebra generated by the projections to the $0,\dots,n$ coordinates by $\mathcal{F}_n$.
A $\textit{Stopping time}$ $\tau :\Omega \to \{0\} \cup \mathbb{N}$ is a function for which
$$
\tau^{-1}\{n\} \in \mathcal{F}_n \text{ for all } n\geq 0.
$$
Question: On the bottom of page 89 of the book (or page 22 of the notes) author defines the $\sigma$-algebra $\mathcal{F}_\tau$ and claims that $X_\tau$ is measurable with respect to this algebra. What is the symbol $X_\tau$? He hasn't defined it before. Am I blind?
I also look at the book of Gut and there's the same symbol on page 495 without a definition.
Best Answer
When we write $X_\tau$, we are indexing into the sequence $X_0, X_1, X_2, \dots$ (of random variables: the states of the Markov chain at time $0, 1, 2, \dots$) by the random variable $\tau$.
Very formally (more formally than anyone writes) $X_\tau$ is the random variable whose value at an outcome $\omega \in \Omega$ is $X_{\tau(\omega)}(\omega)$.