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  "body": "\n## Statement\n\nLet $\\mu$ be a [[complex measure|NY6TYHRGMS]] on the Borel sets of $[0, 2\\pi)$, viewed as the unit circle.\nFor $n \\in \\Z$ let $\\hat\\mu(n) = \\int e^{-int} \\, d\\mu(t)$.\nAssume that $\\hat\\mu(n) = 0$ for every integer $n \u003c 0$.\nThen $\\mu$ is [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].\nMoreover $d\\mu = h \\, dt$ for an integrable function $h$ whose [[Poisson integral|TCYSWCTYYM]] is holomorphic on the unit disc and belongs to the [[Hardy space|PNNNFJVF6W]] $H^1$.\n\n## Remarks\n\nProof idea: the Poisson integral of $\\mu$ is $F(z) = \\frac{1}{2\\pi} \\sum_{n \\ge 0} \\hat\\mu(n) z^n$, a holomorphic function.\nIts integrals $\\int \\abs{F(r e^{it})} \\, dt$ are bounded by the total variation of $\\mu$, so $F \\in H^1$.\nA function in $H^1$ has boundary values $F^*$ with $F(r e^{it}) \\to F^*(e^{it})$ in $L^1$, a fact that uses the [[inner-outer factorization|X6ZQPAQF76]].\nHence the measures $F(r e^{it}) \\, dt$ converge both to $\\mu$ and to $F^* \\, dt$, and $\\mu = F^* \\, dt$.\n\nExample: if $\\mu$ is a point mass at $t = 0$, then $\\hat\\mu(n) = 1$ for all $n$, so the hypothesis fails, as it must since a point mass is not absolutely continuous.\n\nA consequence: if $\\Omega$ is a domain bounded by a rectifiable Jordan curve, then harmonic measure on $\\partial\\Omega$ and arc length have the same null sets.\n\n## Explanations\n\n### L1 human\n\nA measure on the circle has Fourier coefficients, one for each integer.\nIf all the coefficients with negative index vanish, the measure is \"analytic\".\nThe theorem says that an analytic measure cannot be concentrated on a small set: it must have a density.\nSo analytic measures are much more regular than general measures.\n\n### L2 human\n\nThe theorem identifies the analytic measures on the circle with the boundary values of $H^1$ functions.\nIt is a key step in the theory of [[Hardy spaces|PNNNFJVF6W]] and in the study of boundary behaviour of conformal maps.\nIts consequence for rectifiable Jordan domains links the [[harmonic measure|CFD7BPM51W]] with arc length.\n",
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