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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $p_1, \\ldots, p_n \\in [0, 1]$.\nLet $\\Omega = \\set{0, 1}^n$ with the product [[probability measure|059TJN0NZ1]] $\\mu$ under which the coordinates are independent and the $i$-th coordinate equals $1$ with probability $p_i$.\nOrder $\\Omega$ coordinatewise: $x \\le y$ when $x_i \\le y_i$ for all $i$.\nLet $f, g \\colon \\Omega \\to \\R$ be [[increasing|0A7KV2WB7A]] functions, that is, $x \\le y$ implies $f(x) \\le f(y)$ and $g(x) \\le g(y)$.\nThen\n$$\\mathbb{E}[f g] \\ge \\mathbb{E}[f] \\, \\mathbb{E}[g],$$\nwhere $\\mathbb{E}$ is the [[expected value|0WQNRCNY3M]] with respect to $\\mu$.\n\n## Remarks\n\nThe proof is by induction on $n$.\nFor $n = 1$, the inequality is equivalent to $p_1 (1 - p_1) (f(1) - f(0)) (g(1) - g(0)) \\ge 0$, which holds because both differences are nonnegative.\nThe induction step conditions on the last coordinate and applies the case $n = 1$ to the conditional expectations, which are again increasing.\n\nFor indicator functions of increasing events $A$ and $B$, the inequality reads $\\mu(A \\cap B) \\ge \\mu(A) \\, \\mu(B)$: increasing events are positively correlated.\nFor example, in the [[Erdős–Rényi model|6E83YKANV3]] $G(n, p)$ the events \"the graph is connected\" and \"the graph has a triangle\" are increasing, so they are positively correlated.\nIf $f$ is increasing and $g$ is decreasing, the inequality is reversed.\nThe product structure of $\\mu$ matters.\nFor the measure giving probability $1/2$ to each of $(1, 0)$ and $(0, 1)$, the increasing events $\\set{x_1 = 1}$ and $\\set{x_2 = 1}$ have probability $1/2$ each but are disjoint.\n\n## Explanations\n\n### L1 human\n\nUnder independent coin flips, two \"more is better\" quantities tend to go up together.\nIf knowing that some coins show heads makes $f$ larger, it also makes $g$ larger, so the average of the product is at least the product of the averages.\n\n### L2 human\n\nThe Harris inequality is the case of a product measure in the FKG inequality, which allows any measure on a distributive lattice satisfying the log-supermodularity condition.\nBoth follow from the [[Ahlswede–Daykin inequality|4085B8A528]].\nFor the uniform measure on subsets of a finite set, it is Kleitman's lemma: two up-sets $\\mathcal{A}, \\mathcal{B}$ of subsets of $\\set{1, \\ldots, n}$ satisfy $2^n |\\mathcal{A} \\cap \\mathcal{B}| \\ge |\\mathcal{A}| \\, |\\mathcal{B}|$.\nIt is a standard tool in percolation theory and random graphs, often together with the [[Janson inequality|J3CT5D7FSZ]], which gives a bound in the other direction.\n",
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