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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mu$ be a finite [[complex measure|NY6TYHRGMS]] on the [[Borel sets|0WQXY32Q8D]] of $\\R^n$.\nThe *Fourier–Stieltjes transform* of $\\mu$ is the function $\\hat\\mu \\colon \\R^n \\to \\C$ given by\n$$\\hat\\mu(\\xi) = \\int_{\\R^n} e^{-2\\pi i x \\cdot \\xi}\\, d\\mu(x), \\qquad \\xi \\in \\R^n.$$\n\n## Remarks\n\nThe integral exists because the integrand has modulus $1$ and $\\mu$ has finite total variation $\\norm{\\mu}$.\nSo $\\abs{\\hat\\mu(\\xi)} \\le \\norm{\\mu}$, and $\\hat\\mu$ is uniformly continuous on $\\R^n$.\n\nExample: the Dirac measure $\\delta_a$ at a point $a \\in \\R^n$ is a finite measure, and $\\hat\\delta_a(\\xi) = e^{-2\\pi i a \\cdot \\xi}$; in particular $\\hat\\delta_0 = 1$.\n\nExample: if $d\\mu = f\\, dx$ with $f \\in L^1(\\R^n)$, then $\\hat\\mu$ is the usual [[Fourier transform|DPVM34ADZV]] $\\hat f$.\n\nNon-example: Lebesgue measure on $\\R$ is not finite, and $\\int_\\R e^{-2\\pi i x \\xi}\\, dx$ diverges for every $\\xi$; its transform exists only as a tempered distribution.\n\nUnlike the transform of an integrable function, $\\hat\\mu$ need not tend to $0$ at infinity, as the example $\\delta_0$ shows.\nThe transform is injective: $\\hat\\mu = 0$ implies $\\mu = 0$.\nIt turns convolution of measures into products, $\\widehat{\\mu * \\nu} = \\hat\\mu\\, \\hat\\nu$.\nFor a probability measure $\\mu$ on $\\R$, the characteristic function $\\varphi(t) = \\int e^{itx}\\, d\\mu(x)$ equals $\\hat\\mu(-t / (2\\pi))$.\nOn $\\R$, writing $\\mu$ through a function $F$ of bounded variation, the transform is the Stieltjes integral $\\int e^{-2\\pi i x \\xi}\\, dF(x)$.\n",
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