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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] with its [[Borel σ-algebra|0WQXY32Q8D]] $\\mathcal{B}$.\nLet $\\mathcal{P}(X)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $X$.\nFor $A \\subseteq X$ and $\\varepsilon \u003e 0$, let $A^\\varepsilon = \\set{x \\in X : d(x, a) \u003c \\varepsilon \\text{ for some } a \\in A}$.\nThe *Lévy–Prokhorov metric* is the map $\\pi \\colon \\mathcal{P}(X) \\times \\mathcal{P}(X) \\to [0, 1]$ given by\n$$\\pi(\\mu, \\nu) = \\inf \\set{\\varepsilon \u003e 0 : \\mu(A) \\le \\nu(A^\\varepsilon) + \\varepsilon \\text{ and } \\nu(A) \\le \\mu(A^\\varepsilon) + \\varepsilon \\text{ for all } A \\in \\mathcal{B}}.$$\nThe value is an [[infimum|0QF1MZ9FRG]].\n\n## Remarks\n\nThe function $\\pi$ is a metric on $\\mathcal{P}(X)$, and $\\pi \\le 1$ because every $\\varepsilon \\ge 1$ satisfies both inequalities.\nIf $X$ is separable, then $\\pi(\\mu_n, \\mu) \\to 0$ if and only if $\\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\\mu$.\nIf $X$ is separable and complete, then so is $(\\mathcal{P}(X), \\pi)$.\n\nExample: let $\\delta_x$ be the [[Dirac measure|0ACRGYNHSR]] at $x$, and let $x, y \\in X$.\nIf $\\varepsilon \u003e d(x, y)$, then every Borel set $A$ containing $x$ has $y \\in A^\\varepsilon$, so $\\delta_x(A) \\le \\delta_y(A^\\varepsilon)$, and symmetrically.\nHence $\\pi(\\delta_x, \\delta_y) = \\min(d(x, y), 1)$.\n\nNon-example: for $\\varepsilon \u003c \\min(d(x, y), 1)$ the value $\\varepsilon$ is not admissible.\nTake $A = \\set{x}$: then $y \\notin A^\\varepsilon$, and $1 \\le 0 + \\varepsilon$ fails.\n\nOn $\\R$, the points $1/n$ and $0$ give $\\pi(\\delta_{1/n}, \\delta_0) = 1/n$, while the total variation distance stays equal to $1$.\n\n## Explanations\n\n### L1 human\n\nThe metric allows an error of $\\varepsilon$ both in position and in mass.\nIts smallest admissible $\\varepsilon$ measures how far apart the two measures are.\n",
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