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  "body": "\n## Problem\n\nLet $K$ be a closed subset of $\\R$ and let $(m_n)_{n \\ge 0}$ be a [[sequence|0G9EKM0E15]] of real numbers.\n\n1. Existence: decide whether there is a positive Borel [[measure|PR11BZQQPG]] $\\mu$ on $\\R$ with support in $K$ and the following property. Every $x^n$ is $\\mu$-integrable and $m_n = \\int x^n \\, d\\mu(x)$ for all $n \\ge 0$.\n2. Uniqueness: when such a measure exists, decide whether it is the only one.\n\nThe numbers $m_n$ are then the [[moments|01XAHPADBA]] of $\\mu$.\n\n## Remarks\n\nThe classical cases are named after the set $K$.\nFor $K = \\R$ it is the [[Hamburger moment problem|TBNCFXTMMW]], for $K = [0, \\infty)$ the [[Stieltjes moment problem|Y9KH8P9JC0]], and for $K = [0, 1]$ the [[Hausdorff moment problem|G7VG61YD4J]].\nFor these three cases the existence question is solved by positivity conditions on Hankel matrices built from the moments, or on differences of the sequence for $[0, 1]$.\nFor compact $K$ the measure is unique when it exists, since polynomials are dense in the continuous functions on $K$.\nFor unbounded $K$ uniqueness can fail; the log-normal distribution is the standard example.\nMany variants are studied: finitely many moments (the truncated problem), measures on $\\R^d$, and the trigonometric problem on the circle.\n\n## Explanations\n\n### L1 human\n\nThe moments of a mass distribution are its total mass, its average position, its average squared position, and so on.\nThe moment problem runs the other way: given a list of numbers, is there a distribution with exactly these moments, and is there only one?\nThe answer depends on where the mass may sit, on a bounded interval, a half-line or the whole line.\nIt matters in probability, where one often knows the moments of a random variable and wants to identify its law.\n",
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