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  "body": "\n## Statement\n\nLet $L$ be a finite [[distributive lattice|0HM0HQCK6N]] with join $\\vee$ and meet $\\wedge$.\nLet $\\mu \\colon L \\to [0, \\infty)$ be log-supermodular, that is, $\\mu(x \\vee y) \\, \\mu(x \\wedge y) \\ge \\mu(x) \\, \\mu(y)$ for all $x, y \\in L$.\nLet $f, g \\colon L \\to \\R$ be [[increasing|0A7KV2WB7A]]: $x \\le y$ implies $f(x) \\le f(y)$ and $g(x) \\le g(y)$.\nThen\n$$\\Big( \\sum_{x \\in L} f(x) g(x) \\mu(x) \\Big) \\Big( \\sum_{x \\in L} \\mu(x) \\Big) \\ge \\Big( \\sum_{x \\in L} f(x) \\mu(x) \\Big) \\Big( \\sum_{x \\in L} g(x) \\mu(x) \\Big).$$\n\n## Remarks\n\nWhen $\\mu$ is a probability measure, the inequality reads $\\mathbb{E}[f g] \\ge \\mathbb{E}[f] \\, \\mathbb{E}[g]$: increasing functions are positively correlated.\nIt follows from the [[Ahlswede–Daykin inequality|4085B8A528]].\nAdding a constant to $f$ or to $g$ does not change the difference of the two sides, so one may assume $f, g \\ge 0$.\nThen the four functions $f \\mu$, $g \\mu$, $f g \\mu$ and $\\mu$ satisfy $f(x) \\mu(x) \\, g(y) \\mu(y) \\le f(x \\vee y) g(x \\vee y) \\mu(x \\vee y) \\, \\mu(x \\wedge y)$, by monotonicity and log-supermodularity, and Ahlswede–Daykin with both sets equal to $L$ gives the result.\n\nExample: a product measure on the lattice of subsets of $\\set{1, \\ldots, n}$ satisfies the hypothesis with equality, and the theorem becomes the [[Harris inequality|W1YH3YD3S4]].\nIf $f$ is increasing and $g$ is decreasing, the inequality reverses.\nLog-supermodularity cannot be dropped.\nOn $\\set{0, 1}^2$, the measure giving mass $1/2$ to each of $(1, 0)$ and $(0, 1)$ is not log-supermodular.\nFor it, the increasing indicators of $\\set{x_1 = 1}$ and $\\set{x_2 = 1}$ are negatively correlated.\n\n## Explanations\n\n### L1 human\n\nImagine random configurations in which \"more\" of one thing tends to come with \"more\" of another.\nThe FKG inequality makes this precise: if the weights favour combining configurations, then any two increasing quantities are positively correlated.\nFor instance, in a random graph, being connected and containing a triangle tend to happen together.\n\n### L2 human\n\nThe inequality was proved by Fortuin, Kasteleyn and Ginibre for models of statistical mechanics such as the Ising model with ferromagnetic interactions, whose Gibbs measures are log-supermodular.\nIn combinatorics and probability it is used to compare probabilities of monotone events in random graphs and percolation.\nThe Harris inequality is its product-measure case, and the Holley inequality is a two-measure generalization.\nThe XYZ inequality for linear extensions of posets is proved with it.\n",
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