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  "body": "\n## Statement\n\nLet $m \\ge 1$ and let $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\R$ with finite [[moments|01XAHPADBA]] $m_k = \\int_\\R x^k \\, d\\mu(x)$ for $0 \\le k \\le 2m - 1$.\nAssume that the [[Hankel matrix|PQ1D8FQB3E]] $(m_{i+j})_{i,j=0}^{m-1}$ is [[positive definite|0KX66CZMBB]].\nLet $P_m$ be the monic polynomial of degree $m$ that is [[orthogonal|8TXASARQN1]] to all polynomials of degree less than $m$ in $L^2(\\mu)$.\nThen $P_m$ has $m$ simple real zeros $\\xi_1 \u003c \\xi_2 \u003c \\cdots \u003c \\xi_m$.\nLet $\\rho_1, \\ldots, \\rho_m \u003e 0$ be the Christoffel numbers, that is, the unique numbers with $\\int_\\R p \\, d\\mu = \\sum_{j=1}^{m} \\rho_j \\, p(\\xi_j)$ for every real polynomial $p$ of degree at most $2m - 1$.\nLet $\\nu$ be any positive Borel measure on $\\R$ with $\\int_\\R x^k \\, d\\nu(x) = m_k$ for $0 \\le k \\le 2m - 1$.\nThen for each $k$ with $1 \\le k \\le m$,\n$$\\sum_{j=1}^{k-1} \\rho_j \\;\\le\\; \\nu\\bigl((-\\infty, \\xi_k)\\bigr) \\;\\le\\; \\nu\\bigl((-\\infty, \\xi_k]\\bigr) \\;\\le\\; \\sum_{j=1}^{k} \\rho_j.$$\n\n## Remarks\n\nFor the upper bound one builds, by Hermite interpolation, a polynomial $q$ of degree $2m - 2$ with $q(\\xi_j) = 1$ for $j \\le k$, $q(\\xi_j) = 0$ for $j \u003e k$, and $q'(\\xi_j) = 0$ for $j \\ne k$.\nOne shows that $q \\ge 1$ on $(-\\infty, \\xi_k]$ and $q \\ge 0$ on $\\R$.\nThen $\\nu((-\\infty, \\xi_k]) \\le \\int q \\, d\\nu = \\int q \\, d\\mu = \\sum_j \\rho_j q(\\xi_j) = \\sum_{j \\le k} \\rho_j$, since the integrals of $q$ depend only on the moments; the lower bound follows by the reflection $x \\mapsto -x$.\nThe measure $\\sum_j \\rho_j \\delta_{\\xi_j}$ of Gaussian quadrature has the given moments, and it attains every bound.\nIn particular, the mass that any such $\\nu$ puts at the point $\\xi_k$ is at most $\\rho_k$.\nFor example, for $m = 1$ with $m_0 = 1$ and $m_1 = 0$, one gets $\\xi_1 = 0$ and $\\rho_1 = 1$, and the inequalities say only $0 \\le \\nu((-\\infty, 0)) \\le \\nu((-\\infty, 0]) \\le 1$.\nThe positive definiteness of the Hankel matrix holds exactly when $\\mu$ is supported on at least $m$ points; it is needed for $P_m$ and the numbers $\\rho_j$ to exist.\n\n## Explanations\n\n### L1 human\n\nSuppose only the first few moments of a mass distribution on the line are known.\nThen the distribution is not known, but the amount of mass to the left of certain special points is trapped between two explicit numbers.\nThe special points are the zeros of an orthogonal polynomial, and the numbers are sums of the weights of Gaussian quadrature.\n\n### L2 human\n\nThe inequalities say that the distribution functions of all solutions of a truncated moment problem pass through a band around the staircase of the Gaussian quadrature measure.\nThe bound on point masses, $\\nu(\\set{\\xi_k}) \\le \\rho_k$, also follows from the extremal property of the Christoffel function.\n",
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