Blumenthal's zero–one law

In the mathematical theory of probability, Blumenthal's zero–one law,[1] 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 , ∞ )

{\displaystyle [0,\infty )}

starting from deterministic point has also deterministic initial movement.

Suppose that

= (

t

: t ≥ 0 )

{\displaystyle X=(X_{t}:t\geq 0)}

is an adapted right continuous Feller process on a probability space

is constant with probability one.

:= σ (

Then any event in the germ sigma algebra

Suppose that

is an adapted stochastic process on a probability space

is constant with probability one.

has Markov property with respect to the filtration

then any event

Note that every right continuous Feller process on a probability space

has strong Markov property with respect to the filtration