Blumenthal's zero-one law asserts that the germ $\sigma$-algebra $\mathcal{F}_0^+ = \bigcap_{t > 0} \sigma(B_s : s \leq t)$ is trivial: every event in $\mathcal{F}_0^+$ has probability $0$ or $1$. Events in $\mathcal{F}_0^+$ are those determined by the "infinitesimal future" of Brownian motion — they depend on the behaviour of $B$ in the interval $(0, \varepsilon)$ for every $\varepsilon > 0$ but not on the behaviour at any specific time. Blumenthal's law says that such events, despite depending on the process, carry no genuine randomness. The proof is elegant: since $B_0 = 0$ a.s., the full process $(B_t)_{t \geq 0}$ coincides with the increment $(B_t - B_0)_{t \geq 0}$, which by the [Independence from the Germ Field](/theorems/1177) is independent of $\mathcal{F}_0^+$. But every event $A \in \mathcal{F}_0^+$ is also measurable with respect to $\sigma(B_t : t \geq 0) = \sigma(B_t - B_0 : t \geq 0)$, so $A$ is independent of itself: $\mathbb{P}(A) = \mathbb{P}(A \cap A) = \mathbb{P}(A)^2$, forcing $\mathbb{P}(A) \in \{0, 1\}$. Blumenthal's law has immediate and powerful consequences for the local behaviour of Brownian motion near $t = 0$. Events such as $\{\exists \, \varepsilon > 0 : B_t > 0 \; \forall t \in (0, \varepsilon)\}$ or $\{\limsup_{t \downarrow 0} B_t / \sqrt{2t \log \log(1/t)} = 1\}$ belong to $\mathcal{F}_0^+$ and therefore have probability $0$ or $1$. The law is the key input for the [Immediate Return to Zero](/theorems/1179) result, which shows that Brownian motion immediately becomes positive, immediately becomes negative, and immediately returns to zero after time $0$ — all with probability $1$.