I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$.
Let $\mathcal{F}_t^+ = \cap_{s > t} \mathcal{F}_s^0$ and $\mathcal{F}_t^0 = \sigma(B_s,s\le t)$ where $B_t$ is Brownian motion. Also let $P_x$ denote the usual measure on $\mathcal{C}[0,\infty)$ making $B_t(\omega) = \omega(t)$ a Brownian motion starting at $x$.
Blumenthal's 0-1 law: If $A \in \mathcal{F}_0^+$ then for all $x \in \mathbb{R}$, $P_x(A) \in \{0,1\}$.
I have no trouble understanding this proof, but I'm not sure I understand why this is surprising/significant.
Obviously, $\mathcal{F}_0^0$ is trivial. Is there any intuitive way to explain why we might not expect $\mathcal{F}_0^+$ to be trivial as well? Even though $\mathcal{F}_0^+$ seems to involve a tiny bit of information into the future past $t=0$, since Brownian motion is continuous it would seem that we ought to expect no difference between the two fields.