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  "body": "\n## Definition\n\nLet $\\mu$ be a [[probability measure|059TJN0NZ1]] on a measurable space $(E, \\mathcal{E})$ and $\\nu$ a probability measure on a measurable space $(F, \\mathcal{F})$.\nA *coupling* of $\\mu$ and $\\nu$ is a probability measure $\\gamma$ on the product space $(E \\times F, \\mathcal{E} \\otimes \\mathcal{F})$ whose [[marginals|0MXT6PRZZ6]] are $\\mu$ and $\\nu$:\n$$\\gamma(A \\times F) = \\mu(A) \\quad \\text{and} \\quad \\gamma(E \\times B) = \\nu(B) \\qquad \\text{for all } A \\in \\mathcal{E}, \\ B \\in \\mathcal{F}.$$\nEquivalently, a coupling is a pair of [[random variables|0PNRKMRC55]] $(X, Y)$ on one [[probability space|0NFPJVJJHX]] with $X$ of law $\\mu$ and $Y$ of law $\\nu$.\nThe joint law of such a pair is a coupling in the first sense.\n\n## Remarks\n\nExample: the [[product measure|77PV0ZAVXT]] $\\mu \\otimes \\nu$ is a coupling, the independent coupling. Indeed $(\\mu \\otimes \\nu)(A \\times F) = \\mu(A) \\nu(F) = \\mu(A)$, and in the same way for the second marginal.\nExample: for $0 \\le p \\le q \\le 1$, let $U$ be uniform on $[0, 1]$, $X = \\mathbf{1}_{\\set{U \u003c p}}$ and $Y = \\mathbf{1}_{\\set{U \u003c q}}$. Then $X$ and $Y$ are [[Bernoulli|0386Y0Z7Y7]] with parameters $p$ and $q$, and $X \\le Y$ everywhere.\nNon-example: on $\\set{0, 1}^2$, the point mass $\\delta_{(0, 0)}$ is not a coupling of two Bernoulli laws with parameter $\\frac{1}{2}$, because its first marginal is $\\delta_0$.\nIf $E = F$ and the diagonal is measurable, every coupling satisfies the coupling inequality $d_{TV}(\\mu, \\nu) \\le P(X \\ne Y)$, where $d_{TV}$ is the total variation distance.\n",
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