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I am reading the book Brownian Motion by Yuval peres and Peter Morters. I am having trouble understanding the following remark. In perticular, How the set on the right equals the set on the left. The remark is as follows,

Suppose $H$ is a closed set, for example a singleton. Then the first hitting time $T=\inf \{t \geqslant 0: B(t) \in H\}$ of the set $H$ is a stopping time with respect to $\left(\mathcal{F}^{0}(t): t \geqslant 0\right)$. Indeed, we note that $$ \{T \leqslant t\}=\bigcap_{n=1}^{\infty} \bigcup_{s \in \mathbb{Q} \cap(0, t)} \bigcup_{x \in \mathbb{Q}^{d} \cap H}\left\{B(s) \in \mathcal{B}\left(x, \frac{1}{n}\right)\right\} \in \mathcal{F}^{0}(t) . $$

Here, $\mathcal{F}^{0}(t) = \sigma\left(B(s): 0 \leq s \leq t \right)$

I don't get why it is true because, if H a closed set which does not intersect with $\mathbb{Q}^d$ then the intersection on RHS would be just empty.

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    $\begingroup$ You are right. The identity is not correct. $\endgroup$ Commented Jun 7, 2022 at 7:17

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As stated in the comments, the identity is not correct as written. The argument should just be that there exists a countable subset $A \subseteq H$ which is dense in $H$, and then $$ \{T \leqslant t\}=\bigcap_{n=1}^{\infty} \bigcup_{s \in \mathbb{Q} \cap(0, t)} \bigcup_{x \in A}\left\{B(s) \in \mathcal{B}\left(x, \frac{1}{n}\right)\right\} \in \mathcal{F}^{0}(t) . $$

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  • $\begingroup$ Ah, this makes sense, Thanks! So intuitively we are just checking if for some s whether the Brownian motion gets arbitrarily close to an element of $A$ (and consequently and equivalently to the set $H$). And conclude with continuity of $B$ and closedness $H$. $\endgroup$
    – Mathaddict
    Commented Jun 7, 2022 at 16:50
  • $\begingroup$ @Mathaddict Yes, exactly! The continuity of $B$ allows us to only check rational $s$, and closedness of $H$ lets us only check that we get arbitrarily close to $H$. $\endgroup$ Commented Jun 7, 2022 at 19:22
  • $\begingroup$ Thank you. I get it now. $\endgroup$
    – Mathaddict
    Commented Jun 8, 2022 at 5:14

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