I'm trying to get a sense of what the $\sigma$-algebras in a filtration actually contain. More specifically, suppose $W_t$ is standard $\mathbb{R}$-Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ where $(\mathcal{F}_t)_{t \geq 0}$ is the natural filtration given by $W_t$. Now $W_t$ is a Normal random variable, which has support on $(-\infty, \infty)$, so in my mind it seems reasonable then that $\mathcal{F}_t=\mathcal{B}(\mathbb{R})$, the Borel $\sigma$-algebra over $\mathbb{R}$ (since I can measure the probability of $W_t$ essentially being in any "nice" subset of $\mathbb{R}$).
However, since a filtration describes a "flow of information", it clearly is not the case that $\mathcal{F}_t=\mathcal{B}(\mathbb{R})$ for all $t \geq 0$, since then is nothing gained from "being at $t$ rather than $s < t$".
On the other hand, if I were to simulate a sample path of $W_t$, clearly if $W_2=5$, I can sample $W_6$ from $N(5, 1)$, so clearly I have the "information" that I should "start at $5$".
Any insight is appreciated!