On the filtration generated by a continuous stochastic process

Question

Let $(X_t)$ be a stochastic process with sample paths that are $a.s$ continuous (for example, the Brownian motion) defined on some probability space.

I feel that it must be true that $\sigma(X_t : 0 \leq t \leq r)= \sigma(X_t : 0 \leq t \leq r, t \in \mathbb Q)$ because of the $a.s$ continuity of $X_t$. Is it true ? How can we show it ?

Answer 1

Let $\mathcal{F}_r:=\sigma(X_t : 0 \leq t \leq r, t \in \mathbb Q)$ and take $r\notin \mathbb{Q}$. The r.v. $X_r$ is not necessarily $\mathcal{F}_r$-measurable unless the latter is augmented with all the $\mathsf{P}$-null sets.