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  "body": "\n## Example\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$.\nIn 1916 Menshov constructed a closed set $E \\subset \\mathbb{T}$ of Lebesgue measure zero that is a set of multiplicity.\nSo there is a [[trigonometric series|FZK2M8KGXQ]] $\\sum_{n \\in \\Z} c_n e^{inx}$, with coefficients not all $0$, whose symmetric partial sums converge to $0$ at every point outside $E$.\n\nThe construction gives a nonzero positive [[Borel measure|DYR1QFNGNJ]] $\\mu$ supported on a perfect [[null set|VQQM4Y550T]] $E$ whose Fourier–Stieltjes coefficients $\\hat\\mu(n) = \\frac{1}{2\\pi} \\int e^{-inx} \\, d\\mu(x)$ tend to $0$ as $\\abs{n} \\to \\infty$.\nThe series $\\sum_n \\hat\\mu(n) e^{inx}$ then converges to $0$ at every point outside the support of $\\mu$.\nThis follows from Riemann's principle of localization for series with coefficients tending to $0$.\nIts coefficients are not all $0$, since $\\hat\\mu(0) = \\mu(\\mathbb{T}) / 2\\pi \u003e 0$.\nHence $E$ is not a [[set of uniqueness|K7CNWXF9SN]].\n\n## Remarks\n\nThe example shows that measure zero is necessary but not sufficient for a closed set to be a set of uniqueness.\nA measure whose Fourier coefficients tend to $0$ is called a Rajchman measure, and its closed support is always a set of multiplicity by the same argument.\nWhether a set of measure zero is a set of uniqueness can depend on fine arithmetic properties of the set.\n\n## Explanations\n\n### L1 human\n\nCantor showed that a trigonometric series that is $0$ everywhere must be the zero series, and later this was extended to series that are $0$ outside a countable set.\nOne might guess that any exceptional set of total length zero is harmless.\nMenshov showed that this guess is wrong: he built a set of length zero and a nonzero series that is $0$ everywhere outside it.\nThe series hides all of its non-zero behaviour on a tiny, dust-like set.\n",
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