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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[independent|0XCE070V8B]] random variables on a probability space $(\\Omega, \\mathcal{F}, P)$.\nLet $\\mathcal{T} = \\bigcap_{n=1}^{\\infty} \\sigma(X_n, X_{n+1}, \\dots)$ be the tail σ-algebra, where $\\sigma(X_n, X_{n+1}, \\dots)$ is the smallest [[σ-algebra|04WSW8T3SY]] that makes $X_m$ measurable for all $m \\ge n$.\nThen $P(A) \\in \\{0, 1\\}$ for each $A \\in \\mathcal{T}$.\n\n## Remarks\n\nAn event is in $\\mathcal{T}$ if changing finitely many of the $X_n$ cannot change whether it occurs.\nExamples of tail events: \"the series $\\sum_n X_n$ converges\" and \"$\\limsup_n (X_1 + \\dots + X_n)/n \\le c$\" for a real $c$.\nProof idea: $\\mathcal{T}$ is [[independent|09T8ZHST1V]] of $\\sigma(X_1, \\dots, X_n)$ for each $n$, hence of $\\sigma(X_1, X_2, \\dots)$, which contains $\\mathcal{T}$. So each $A \\in \\mathcal{T}$ is independent of itself, and $P(A) = P(A)^2$.\nThe theorem says that a tail event is almost sure or almost impossible. It does not say which of the two cases holds.\n\n## History\n\nThe theorem is named after Andrey Kolmogorov.\n\n## Explanations\n\n### L1 human\n\nSome events depend only on the \"far end\" of an infinite sequence of independent trials, such as \"the averages converge\".\nFor such an event, the probability cannot be $1/2$ or any value strictly between $0$ and $1$: it is always $0$ or $1$.\n",
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