How to picture a filtration generated by a Brownian motion?

Question

I know what filtrations generated by discrete random variables are, but I can't apply it to Brownian motions. Please help.

Answer 1

One view of the matter is this. Suppose $(\Omega,\mathcal F,\Bbb P)$ is a probability space on which is defined a Brownian motion $(B_t)_{t\ge 0}$, and let $\mathcal F_t:=\sigma\{B_s:0\le s\le t\}$. Let $A$ be an element of $\mathcal F_t$ for some fixed $t>0$. If now $\omega$ and $\omega'$ are two elements of $\Omega$ such that $B_s(\omega) = B_s(\omega')$ for all $s\in[0,t]$, then $\omega\in A$ if and only if $\omega'\in A$.