Blumenthal zero-one law (${\mathcal{F}}_{0+}={\bigcap }_{t>0}{\mathcal{F}}_t\text{ is trivial}$)

For the completed natural Brownian filtration, every event in the germ sigma-field of Brownian motion has probability zero or one. Indeed, an event measurable at every positive time is independent of all increments after each such time. Letting that time decrease to zero and using path continuity makes the event independent of the entire Brownian path sigma-field. Since it is itself measurable in that sigma-field, its probability equals its square.