= Kolmogorov's zero–one law
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= Kolmogorov zero-one law
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For mutually <independent random variables>, every <tail event> has $\mathbb P(A)\in\{0,1\}$. Identical <probability distributions> are unnecessary. The <tail sigma-algebra> is trivial in <probability>: its <events> have <probability> zero or one, though it can contain nonempty proper <sets>.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[Probability: Theory and Examples (fifth edition), §2.5, Theorem 2.5.3, printed pp. 81–82]{ref}
For a proof, set $\mathcal F_m=\sigma(X_1,\ldots,X_m)$ and $\mathcal F_X=\sigma(X_1,X_2,\ldots)$. <Independence extended from generating pi-systems> makes $\mathcal F_m$ independent of $\sigma(X_{m+1},X_{m+2},\ldots)$, hence of each <tail event> $A$. The increasing <union> $\bigcup_m\mathcal F_m$ is a generating <pi-system> for $\mathcal F_X$. For fixed $A$, the <events> $B$ satisfying $\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)$ form a <Dynkin system>. The <pi-lambda theorem> extends this equality to all $B\in\mathcal F_X$. Taking $B=A$ gives
$$
\mathbb P(A)=\mathbb P(A\cap A)=\mathbb P(A)^2,
\qquad \mathbb P(A)(1-\mathbb P(A))=0.
$$
Thus $A$ is independent of itself, forcing the two stated values.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[Probability: Theory and Examples (fifth edition), proof of Theorem 2.5.3, steps (a) and (b), printed pp. 81–82]{ref}
The useful feature is the absence of intermediate <probabilities> for infinite properties that ignore every finite prefix. It reduces some calculations to showing that a <tail event> has positive <probability>, or that its <complement> does. The theorem supplies no finite observation deadline, and an extra argument such as the calculation in <infinitely many heads as a tail event> is needed to decide which value occurs.
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