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  "body": "\n## Definition\n\nLet $K$ be a closed subset of $\\R$.\nLet $\\mu$ be a positive Borel [[measure|PR11BZQQPG]] on $\\R$ with support in $K$, such that every [[moment|01XAHPADBA]] $m_n = \\int x^n \\, d\\mu(x)$, $n \\ge 0$, is finite.\nThe [[moment problem|CS7TWHY7JM]] for $\\mu$ on $K$ is *determinate* if the following holds.\nEvery positive Borel measure $\\nu$ with support in $K$ and $\\int x^n \\, d\\nu(x) = m_n$ for all $n \\ge 0$ equals $\\mu$.\n\n## Remarks\n\nExample: the standard normal distribution is determinate on $K = \\R$.\nIts moments are $m_{2n} = (2n - 1)!! \\le (2n)^n$ and $m_{2n+1} = 0$, so $\\sum_n m_{2n}^{-1/(2n)}$ diverges, and Carleman's condition applies.\nExample: every measure with support in a compact set $K$ is determinate on $K$, by the density of polynomials in $C(K)$.\nNon-example: the log-normal distribution with density $f(x) = (x \\sqrt{2\\pi})^{-1} e^{-(\\ln x)^2/2}$ on $(0, \\infty)$ is not determinate, since $f(x)(1 + \\sin(2\\pi \\ln x))$ is a different density with the same moments.\nDeterminacy depends on $K$: a measure on $[0, \\infty)$ can be determinate among measures on $[0, \\infty)$ (Stieltjes determinate) but not among measures on $\\R$ (Hamburger determinate).\n",
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