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  "body": "\n## Example\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] with a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_n)_{n \\ge 0}$, and let $X$ be a real random variable with $E[\\abs{X}] \u003c \\infty$.\nFor $n \\ge 0$ let\n$$M_n = E[X \\mid \\mathcal{F}_n],$$\nthe [[conditional expectation|0YAMJRK04E]] of $X$ given $\\mathcal{F}_n$.\nThen $(M_n)_{n \\ge 0}$ is a [[martingale|0CDVK9RQBH]], the Doob martingale of $X$.\n\nCheck of the three clauses:\n\n1. $M_n$ is $\\mathcal{F}_n$-measurable by the definition of conditional expectation.\n2. $E[\\abs{M_n}] \\le E\\bigl[E[\\abs{X} \\mid \\mathcal{F}_n]\\bigr] = E[\\abs{X}] \u003c \\infty$.\n3. Since $\\mathcal{F}_n \\subseteq \\mathcal{F}_{n+1}$, the tower property gives $E[M_{n+1} \\mid \\mathcal{F}_n] = E\\bigl[E[X \\mid \\mathcal{F}_{n+1}] \\mid \\mathcal{F}_n\\bigr] = E[X \\mid \\mathcal{F}_n] = M_n$.\n\n## Remarks\n\nA Doob martingale is uniformly integrable, and by [[Lévy's upward theorem|W01FD2A8NK]] it converges almost surely and in $L^1$ to $E[X \\mid \\mathcal{F}_\\infty]$, where $\\mathcal{F}_\\infty$ is the σ-algebra generated by all $\\mathcal{F}_n$.\nIn concentration of measure one takes $X = f(Y_1, \\dots, Y_N)$ for independent $Y_i$ and $\\mathcal{F}_n = \\sigma(Y_1, \\dots, Y_n)$.\nThen $M_0 = E[X]$ and $M_N = X$, so $X - E[X]$ is a sum of martingale differences, and [[Azuma's inequality|BDA5TTB1F0]] gives [[McDiarmid's inequality|8APFT1RPM4]].\n\n## Explanations\n\n### L1 human\n\nSuppose a random quantity $X$ is revealed bit by bit over time.\nAt each step, $M_n$ is your best guess of $X$ given what you have seen so far.\nYour guess changes as information arrives, but you cannot predict in which direction it will change next.\nThat is exactly the fair-game property of a martingale.\n",
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