Blumenthal's zero–one law

In the mathematical theory of probability, Blumenthal's zero–one law, named after Robert McCallum Blumenthal, is a statement about the nature of the beginnings of right continuous Feller process. Loosely, it states that any right continuous Feller process on $$[0,\infty)$$ starting from deterministic point has also deterministic initial movement.

Statement
Suppose that $$X=(X_t:t\geq 0)$$ is an adapted right continuous Feller process on a probability space $$(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$$ such that $$X_0$$ is constant with probability one. Let $$\mathcal{F}^X_t:=\sigma(X_s; s\leq t), \mathcal{F}^X_{t^+}:=\bigcap_{s>t}\mathcal{F}^X_s$$. Then any event in the germ sigma algebra $$ \Lambda \in \mathcal{F}^X_{0+}$$ has either $$\mathbb{P}(\Lambda)=0$$ or $$\mathbb{P}(\Lambda)=1.$$

Generalization
Suppose that $$X=(X_t:t\geq 0)$$ is an adapted stochastic process on a probability space $$(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$$ such that $$X_0$$ is constant with probability one. If $$X$$ has Markov property with respect to the filtration $$\{\mathcal{F}_{t^+}\}_{t\geq 0}$$ then any event $$ \Lambda \in \mathcal{F}^X_{0+}$$ has either $$\mathbb{P}(\Lambda)=0$$ or $$\mathbb{P}(\Lambda)=1.$$ Note that every right continuous Feller process on a probability space $$(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$$  has strong Markov property with respect to the filtration $$\{\mathcal{F}_{t^+}\}_{t\geq 0}$$.