Consider a homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with discrete state space $S$. Now consider the map $$T_{ij}=\text{min}\{n\in\mathbb N\,:\, X_n=j\mid X_0=i\}$$ where $T_{ij}$ is defined to be $+\infty$, when doesn't exist any $n$ such that $X_n=j$. I don't understand why this map $T_{ij}$ is a random variable. How to prove the measurability?
Many thanks in advance.
You know that $\{ X_k = r \}$ is measurable, so you can write $\{ T_{ij} = n \}$ as $\{ X_1 \neq j,X_2 \neq j,\dots,X_{n-1} \neq j,X_n = j \}$, then break that up into $\{ X_1 \neq j \} \cap \dots \cap \{ X_{n-1} \neq j \} \cap \{ X_n = j \}$. Then each of the first $n-1$ terms can be written as a finite union of sets of the form $\{ X_k = r \}$, each of which is measurable.