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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_n)_{n \\ge 0}$, and let $\\mathcal{F}_\\infty$ be the [[σ-algebra|0SV8K0TDGY]] generated by $\\bigcup_{n} \\mathcal{F}_n$.\nLet $X$ be a real random variable with $E[\\abs{X}] \u003c \\infty$.\nThen\n$$E[X \\mid \\mathcal{F}_n] \\to E[X \\mid \\mathcal{F}_\\infty] \\quad \\text{as } n \\to \\infty,$$\n[[almost surely|0AAXWGT5TD]] and in $L^1$, where $E[\\cdot \\mid \\cdot]$ is the [[conditional expectation|0YAMJRK04E]].\n\n## Remarks\n\nThe sequence $M_n = E[X \\mid \\mathcal{F}_n]$ is the [[Doob martingale|Z57CG6W9S6]] of $X$, and it is uniformly integrable.\nBy [[Doob's martingale convergence theorems|MRXBC4ZNRR]] it converges almost surely and in $L^1$ to some $M_\\infty$, which is $\\mathcal{F}_\\infty$-measurable.\nFor $A \\in \\mathcal{F}_m$ and $n \\ge m$, $E[M_n \\mathbf{1}_A] = E[X \\mathbf{1}_A]$, and the $L^1$ limit gives $E[M_\\infty \\mathbf{1}_A] = E[X \\mathbf{1}_A]$.\nA π–λ argument extends this to all $A \\in \\mathcal{F}_\\infty$, so $M_\\infty = E[X \\mid \\mathcal{F}_\\infty]$.\nTaking $X = \\mathbf{1}_A$ with $A \\in \\mathcal{F}_\\infty$ gives Lévy's zero–one law: $P(A \\mid \\mathcal{F}_n) \\to \\mathbf{1}_A$ almost surely.\n\n## Explanations\n\n### L1 human\n\nAs you learn more and more, your best guess of a random quantity approaches the best guess based on all the information you will ever have.\nIf all the information together determines the quantity, the guesses converge to the true value.\nFor example, learn the binary digits of a uniform random number one by one.\nYour best guess of any integrable function of the number then converges to its true value almost surely.\n\n### L2 human\n\nThe theorem is a probabilistic counterpart of the Lebesgue differentiation theorem.\nOn $[0, 1)$ with the dyadic filtration, $E[X \\mid \\mathcal{F}_n]$ is the average of $X$ over the dyadic interval of length $2^{-n}$ containing the point.\nLévy's zero–one law follows at once, and it gives Kolmogorov's zero–one law for tail events of independent sequences.\nA companion result, Lévy's downward theorem, treats decreasing families of σ-algebras.\n",
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