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  "body": "\n## Definition\n\nLet $\\gamma$ be a Borel [[probability measure|059TJN0NZ1]] on $\\R$.\nIt is *Gaussian* in two cases.\nThe first case is $\\gamma = \\delta_m$, the [[Dirac measure|0ACRGYNHSR]] at some $m \\in \\R$.\nThe second case is that there are $m \\in \\R$ and $\\sigma \u003e 0$ with\n$$\\gamma(A) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\int_A e^{-(x - m)^2 / (2 \\sigma^2)} \\, dx$$\nfor every Borel set $A \\subseteq \\R$, where $dx$ is [[Lebesgue measure|2Z2QVWZ262]].\nLet $E$ be a real [[separable|0Z2P41SVDP]] [[Banach space|06GZ956657]].\nA Borel probability measure $\\gamma$ on $E$ is *Gaussian* if for every [[continuous linear form|03TM135G0N]] $\\ell \\colon E \\to \\R$ the [[pushforward|3N66TM3TSQ]] $\\ell_* \\gamma$ is a Gaussian measure on $\\R$.\nIt is *centered* if every $\\ell_* \\gamma$ has mean $0$.\n\n## Remarks\n\nIn the second case on $\\R$, $\\gamma$ is the [[normal distribution|05AQE0QR0D]] $N(m, \\sigma^2)$; the Dirac case is the degenerate normal distribution $N(m, 0)$.\nThe *standard* Gaussian measure on $\\R^n$ has density $(2\\pi)^{-n/2} e^{-\\abs{x}^2/2}$.\nSome authors call $\\gamma$ Gaussian only when every $\\ell_* \\gamma$ with $\\ell \\ne 0$ is non-degenerate.\n\nExample: on $E = \\R^2$, the standard Gaussian measure is Gaussian.\nFor $\\ell(x) = a x_1 + b x_2$, the law $\\ell_* \\gamma$ is $N(0, a^2 + b^2)$, since a linear combination of independent standard normal variables is normal.\n\nNon-example: the uniform probability measure on $[-1, 1]$ is not Gaussian.\nIts pushforward under the identity is not a Dirac measure and has no Gaussian density, since it vanishes outside $[-1, 1]$.\n\nWiener measure on the Banach space of continuous functions $\\omega$ on $[0, 1]$ with $\\omega(0) = 0$ is a centered Gaussian measure.\n\n## Explanations\n\n### L1 human\n\nA measure on a vector space is Gaussian when every linear view of it is a bell curve, possibly squeezed to a point.\nThis definition works in any dimension, including infinite dimension.\n",
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