How to prove that the first hitting time is measurable?

Question

I'm reading the definition of first hitting time in a class of martingale:

Assume

• $\left(\Omega, \mathcal{F},(\mathcal{F}_{n})_{n \in \mathbb{N}}, \mathbb P\right)$ is a filtered probability space.

• $(X_{n})_{n \in \mathbb{N}}$ is a stochastic process adapted to the filtration $(\mathcal{F}_{n})_{n \in \mathbb{N}}$.

• $B \subseteq \mathbb{R}$ a Borel set.

We define a map $T_B: \Omega \to \mathbb R$ by $$\forall \omega \in \Omega: T_{B}(\omega) =\inf \left\{n \geqslant 0 \mid X_{n}(\omega) \in B\right\}$$ with the traditional convention that $\inf \varnothing=\infty$. Then $T_{B}$ is a stopping time, called the first hitting time of $B$.

My textbook as well as Wikipedia's page do not prove that $T_B$ is actually measurable. By definition, I must show that $$\forall A \in \mathcal B (\mathbb R) : T_B^{-1} (A) \in \mathcal F$$

But I'm stuck at describing the set $T_B^{-1} (A)$.

How can I proceed to prove that $T_B$ is measurable? Many thanks!

Answer 1

Note that $T_B$ can take only integer values or $\infty$. There is sufficient to look at events $\{T_B = n\}$ for all $n$. $$ \{T_B = n\} = \bigcap_{i=0}^{n-1} \{X_i \not\in B\}\cap \{X_n\in B\} \in \mathcal F_n\subset \mathcal F. $$ There is no need to consider $T_B^{-1} (A)$ since it is equals to $$T_B^{-1} (A) = \{T_B\in A\}=\bigcup_{n\in A\cap\{0,1,2,\ldots\}}\{T_B=n\}$$

Answer 2

To show that $T_B$ is measurable function it suffices to show that the events $\{T_B \gt t\}$ are measurable for all $t$. But $$ \{T_B \gt t\} = \bigcap_{n=0}^{\lfloor t \rfloor}\{X_n \notin B\}, $$ a finite intersection of measurable events.