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  "body": "\n## Statement\n\nLet $E$ be a real [[nuclear space|XC6XPMC2FQ]] that is also a Fréchet space, and let $E'$ be its topological [[dual space|07JBSZE0V4]].\nGive $E'$ the σ-algebra generated by the maps $x \\mapsto x(f)$ for $f \\in E$.\nLet $\\varphi \\colon E \\to \\C$ be a function.\nThen there is a [[probability measure|059TJN0NZ1]] $\\mu$ on $E'$ with\n$$\\varphi(f) = \\int_{E'} e^{i x(f)} \\, d\\mu(x) \\quad \\text{for all } f \\in E$$\nif and only if these three conditions hold:\n\n1. $\\varphi$ is continuous.\n2. $\\varphi$ is positive-definite: $\\sum_{j,k=1}^n c_j \\overline{c_k} \\, \\varphi(f_j - f_k) \\ge 0$ for all $n$, all $f_1, \\dots, f_n \\in E$ and all $c_1, \\dots, c_n \\in \\C$.\n3. $\\varphi(0) = 1$.\n\nWhen $\\mu$ exists, it is unique.\n\n## Remarks\n\nThe function $\\varphi$ is the characteristic functional of $\\mu$, the infinite-dimensional analogue of a [[characteristic function|0V5B6D6BQH]].\nNecessity of the three conditions is direct.\nFor sufficiency, Bochner's theorem on each finite-dimensional quotient gives a [[cylinder set measure|G95YDNFXQN]] on $E'$.\nNuclearity of $E$ is what makes this cylinder set measure countably additive.\n\nExample: on the Schwartz space $E = \\mathcal{S}(\\R^n)$, the function $\\varphi(f) = \\exp\\left(-\\frac12 \\int_{\\R^n} f(t)^2 \\, dt\\right)$ satisfies the three conditions.\nThe measure it gives on the space of tempered distributions $\\mathcal{S}'(\\R^n)$ is white noise.\n\nNuclearity cannot be dropped.\nOn an infinite-dimensional real Hilbert space $H$, the function $\\exp(-\\frac12 \\norm{f}^2)$ is continuous, positive-definite and equal to $1$ at $0$.\nYet there is no probability measure on $H' \\cong H$ with this characteristic functional.\n\n## Explanations\n\n### L1 human\n\nA probability measure on $\\R^n$ is determined by its characteristic function, and Bochner's theorem says which functions occur.\nMinlos's theorem gives the same answer in infinite dimensions, for the dual of a nuclear space.\nSo to build a random distribution, it suffices to write down a suitable characteristic functional.\n\n### L2 human\n\nThe theorem is the standard way to construct Gaussian and other random fields as random elements of $\\mathcal{S}'(\\R^n)$.\nWhite noise, the Gaussian free field and Euclidean quantum fields are built this way.\nIt contrasts with the Hilbert space case, where by Sazonov's theorem continuity in a stronger topology is needed.\n",
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