I would like to know what are the theorems and/or results invoked in this kinds of equalities:
$$ P_x[\tau_r < \infty] = \lim_{R\to\infty}P_x[\tau_{r,R}=\tau_r] $$
where, to give an example rearranging part of the proof of Theorem 25.40 of Probability Theory by A. Klenke (3rd version),
- $\tau_s=\inf\{t>0: W_t=s\}$ and $\tau_{r,R}=\inf\{t>0: W_t\notin G_{r,R}\}$ are stopping times related to a Brownian motion $W$ started in $x\in(r,R)$ and an interval $G_{r,R}:=(r,R)$ with $-\infty<r<0<R<\infty$
My two ideas of what could be involved:
- Monotone Convergence Theorem: I can't figure out what is the sequence of function involved in the second equality. In particular, I do not know how to treat the indicator function to "bring out the limit from the set". For the first equality I do not know what could be a useful result to show that it holds.
$$ E_x[\mathcal{I}_{\{\tau_r<\infty\}}]=E_x[\mathcal{I}_{\{\lim_{R\to\infty}\tau_{r,R}=\tau_r\}}]=\lim_{R\to\infty}E_x[\mathcal{I}_{\{\tau_{r,R}=\tau_r\}}] $$
- Or, define the events $A_R:=\{\tau_{r,R}=\tau_r\}$; see that this is an increasing sequence (how can I prove it?) of events that converge to $A=\{\tau_r<\infty\}$. Then use the lower semi continuity property of the probability measure $P_x[\cdot]$
Please, let me know if more context is needed. Thanks for the help.
Notice that $\{\tau_{r,R}=\tau_r \}=\{\tau_r<\tau_R\}$. As $R\to \infty$ the stopping time $\tau_R$ increases to $\infty$, so the event $\{\tau_r<\tau_R\}$ increases to $\{\tau_r< \infty\}$. This sets up your idea 2.