Let $T=\inf \{t\geq 0: X_t\leq c\}$. Let $X$ be a continuous and adapted process.
Why do we have $$\{T\leq t\}=\bigcap_{n \in \mathbb{N}}\bigcup_{s \in \mathbb{Q}\cap[0,t]}\{X_s\leq c+n^{-1}\} \in \mathcal{F}_t $$?
My reasoning so far:
$\inf\{t\geq 0:X_t\leq c\}\leq t \Leftrightarrow \exists_{s \in [0,t]} X_s\leq c$, and so $\{T\leq t\}=\bigcup_{s \in [0,t]}\{X_s\leq c\}$ The problem now, is that we have an uncountable union of $\mathcal{F}_t$-measurable sets which may result in a non-measurable set.
So, somehow(not very sure how...), we have $\bigcup_{s \in [0,t]}\{X_s\leq c\}=\bigcup_{s \in \mathbb{Q}\cap[0,t]}\{X_s\leq c\}$... this makes it a countable union of measurable sets, but I must be missing something... otherwise, what would be the point of intersecting with the index $n$(this seems to be using continuity)?