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  "body": "\n## Problem\n\nLet $(m_n)_{n \\ge 0}$ be a sequence of real numbers.\nDecide whether there is a positive Borel [[measure|PR11BZQQPG]] $\\mu$ on $[0, \\infty)$ such that every $x^n$ is $\\mu$-integrable and\n$$m_n = \\int_{[0, \\infty)} x^n \\, d\\mu(x) \\quad \\text{for all } n \\ge 0.$$\nThis is the [[moment problem|CS7TWHY7JM]] on the half-line $K = [0, \\infty)$, and the $m_n$ are the [[moments|01XAHPADBA]] of $\\mu$.\n\n## Remarks\n\nThe existence question is solved.\nSuch a measure exists if and only if, for every $n \\ge 0$, both [[Hankel matrices|PQ1D8FQB3E]] $(m_{j+k})_{j,k=0}^{n}$ and $(m_{j+k+1})_{j,k=0}^{n}$ are [[positive semidefinite|01MMVW8CKC]].\nA measure with infinite support exists if and only if all the determinants of these matrices are strictly positive.\nNecessity is direct: for real $c_0, \\ldots, c_n$, the quadratic forms equal $\\int p(x)^2 \\, d\\mu$ and $\\int x \\, p(x)^2 \\, d\\mu$ with $p(x) = \\sum_j c_j x^j$, and $x \\ge 0$ on the support.\nThe solution need not be unique; the log-normal distribution gives a sequence with many solutions.\nA Stieltjes moment sequence can be determinate as a Stieltjes problem but indeterminate as a Hamburger problem.\n\n## Explanations\n\n### L1 human\n\nThe problem asks which lists of numbers are the moments of a mass distribution placed on the nonnegative half-line.\nThe answer is a positivity test on two square tables built from the list.\nThe second table is the first one with the list shifted by one place, and its positivity reflects that the mass sits where $x \\ge 0$.\nStieltjes met this problem in his work on continued fractions.\n",
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