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I was reading in Varadhan's book: probability theory and where I found the following exercise $5.14$. I also came across a similar question: How to prove $\mathcal F_T \subseteq \sigma(\bigcup_n \mathcal F_{T_n})$?

Can we claim that $\Omega\subset\bigcup_{n \in \mathbb{N}}\{\tau_n=\tau\}$? Why ? In case the claim was wrong how can we prove that $\mathcal{F}_{\tau} \subset\left(\bigcup_{n \in \mathbb{N}}\mathcal{F}_{\tau_n}\right)$?

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mathex
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To answer your question: No. Take e.g. the degenerate stopping times $\tau_n=1-1/n$ and $\tau=1$. Then, clearly $\{\tau_n=\tau\}=\emptyset$ for all $n\in\mathbb{N}$ and so their union is again empty.

I'm curious as to why you would need this in order to solve the posted exercise?

Edit: The stated inclusion is correct (with equality, even) if the stopping times take values in $\mathbb{N}$ and $\tau(\omega)<\infty$ for all $\omega\in\Omega$. This is due to the fact that if $\{a_n\}$ is an increasing sequence in $\mathbb{N}$ converging to some $a\in\mathbb{N}$, there must be some $N\in\mathbb{N}$ such that $|a_n-a|<1$ and hence $a_n=a$ for all $n>N$.

muldyr
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  • I am trying to solve the problem, I was wondering if the argument cited in the link is correct or not, but it seems it isn't, any ideas how to prove the inclusion? – mathex Sep 29 '22 at 15:23
  • I realized I may have been too hasty earlier. The inclusion you wrote initially is indeed false as shown by the above example. If, however, you are working in discrete time, i.e. all stopping times taking values in $\mathbb{N}$ (as seems to be the case in the linked solution), then the inclusion holds. I've added an edit to my initial answer to address this. – muldyr Sep 29 '22 at 16:30
  • What if $\tau(w)=\infty$? – mathex Sep 29 '22 at 16:57