Formulation
A more general statement of Kolmogorov's zero-one law holds for sequences of independent σ-algebras. Let (Ω,F,P) be a probability space and let Fn be a sequence of mutually independent σ-algebras contained in F. Let
be the smallest σ-algebra containing Fn, Fn+1, …. Then Kolmogorov's zero-one law asserts that for any event
one has either P(F) = 0 or 1.
The statement of the law in terms of random variables is obtained from the latter by taking each Fn to be the σ-algebra generated by the random variable Xn. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all Xn, but which is independent of any finite number of Xn. That is, a tail event is precisely an element of the intersection .
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