Proof of the existence of minimal stopping times

Question

I'm trying to understand the proof by Itrel Monroe to the existence of minimal stopping times, in particular the following proposition from "On embedding right continuous martingales in Brownian Motion". Proposition 2

I struggle to understand a few points:

1. Monroe consider a partial order "$\leq$" on the set of stopping times having the same law; it is a pointwise or almost sure inequality? I'm used to a.s. relations between random variables but here it isn't specified so I'm not sure.
2. The sequence of stopping times $T_n$ is said to converge to a stopping time $T$ given that the sequence is decreasing; I presume pointwise, is it correct? Again I'm not sure.
3. After saying that the $T_n \to T$ Monroe states that then $W_{T_n} \to W_T$ where $W_t$ is a Brownian motion. Here it seems implicit that there exists a result which guarantees some kind of convergence (what kind?) between stopped processes when the stopping times converges. I have not been able to find something along this line online.

Thank you in advance. It's my first question and English is not my native language so I apologize if I did something wrong.

Answer 1

1. Yes, the ordering here is the pointwise ordering: $T\le S$ means $T(\omega)\le S(\omega)$ for all $\omega$.

2. The extracted chain $\{T_n\}$ decreases pointwise, and the pointwise limit is $T$, another stopping time.

3. Because $\lim_nT_n(\omega)=T(\omega)$ for each $\omega$ and $t\mapsto W_t(\omega)$ is continuous for each $\omega$, you have $\lim_nW_{T_n}(\omega)=\lim_n W_{T_n(\omega)}(\omega)=W_{T(\omega)}(\omega) = W_T(\omega)$ for each $\omega$.