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  "body": "\n## Statement\n\nLet $\\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\\R$, and let $S_\\mu(z) = \\int_\\R \\frac{d\\mu(t)}{t - z}$ be its [[Stieltjes transform|RG1HFAYD98]] for $z \\in \\C \\setminus \\R$.\nThen for all real numbers $a \u003c b$,\n$$\\lim_{\\varepsilon \\to 0^+} \\frac{1}{\\pi} \\int_a^b \\operatorname{Im} S_\\mu(x + i\\varepsilon)\\, dx = \\mu\\big((a, b)\\big) + \\tfrac12 \\mu(\\set{a}) + \\tfrac12 \\mu(\\set{b}).$$\nIn particular, $S_\\mu$ determines $\\mu$.\n\n## Remarks\n\nFor $\\varepsilon \u003e 0$ one has $\\frac{1}{\\pi} \\operatorname{Im} S_\\mu(x + i\\varepsilon) = \\int_\\R P_\\varepsilon(x - t)\\, d\\mu(t)$ with the [[Poisson kernel|TCYSWCTYYM]] $P_\\varepsilon(u) = \\frac{1}{\\pi} \\frac{\\varepsilon}{u^2 + \\varepsilon^2}$ of the upper half-plane.\nBy Fubini, the left side before the limit is $\\int_\\R \\frac{1}{\\pi}\\big(\\arctan\\frac{b - t}{\\varepsilon} - \\arctan\\frac{a - t}{\\varepsilon}\\big)\\, d\\mu(t)$.\nThe integrand is bounded by $1$ and tends to $1$ for $a \u003c t \u003c b$, to $\\frac12$ for $t = a$ or $t = b$, and to $0$ otherwise, so dominated convergence gives the formula.\nIf $\\mu$ has a continuous density $\\rho$ near $x$, then $\\rho(x) = \\lim_{\\varepsilon \\to 0^+} \\frac{1}{\\pi} \\operatorname{Im} S_\\mu(x + i\\varepsilon)$.\n\nExample: for $\\mu = \\delta_0$, $\\frac{1}{\\pi} \\operatorname{Im} S_\\mu(x + i\\varepsilon) = \\frac{1}{\\pi} \\frac{\\varepsilon}{x^2 + \\varepsilon^2}$, which concentrates at $0$ with total mass $1$.\n\n## Explanations\n\n### L1 human\n\nThe Stieltjes transform loses no information: the measure can be recovered from it.\nPush the variable $z$ down to the real axis from above and look at the imaginary part.\nIn the limit it becomes the density of the measure, up to the factor $\\pi$.\n\n### L2 human\n\nThe formula is the reason why the Stieltjes transform is the standard tool to identify a measure.\nIn random matrix theory one computes the limit of the Stieltjes transforms of empirical eigenvalue distributions, and the inversion formula then gives the limiting law, such as the semicircle law.\nIn the moment problem and in the spectral theory of self-adjoint operators the same formula recovers a spectral measure from its resolvent.\n",
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  "msc": [
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