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  "body": "\n## Definition\n\nLet $g \\colon [0, \\infty) \\to \\C$ be a function of [[bounded variation|F34S6P61SW]] on each interval $[0, T]$ with $T \u003e 0$.\nThe *Laplace–Stieltjes transform* of $g$ is the function\n$$\\mathcal{L}^*\\{g\\}(s) = \\int_0^\\infty e^{-s t} \\, dg(t) = \\lim_{T \\to \\infty} \\int_0^T e^{-s t} \\, dg(t),$$\ndefined for each complex $s$ for which the limit exists.\nEach integral $\\int_0^T e^{-s t} \\, dg(t)$ is a [[Riemann–Stieltjes integral|8NMH42GJRK]].\n\n## Remarks\n\nExample: let $a \u003e 0$ and $g(t) = 0$ for $t \u003c a$, $g(t) = 1$ for $t \\ge a$.\nThen $\\int_0^T e^{-s t} \\, dg(t) = e^{-a s}$ for every $T \u003e a$, so $\\mathcal{L}^*\\{g\\}(s) = e^{-a s}$ for every complex $s$.\nIf $g(t) = \\int_0^t f(\\tau) \\, d\\tau$ with $f$ continuous, then $dg = f(t) \\, dt$ and $\\mathcal{L}^*\\{g\\}$ is the [[Laplace transform|05XT5AVJBK]] of $f$.\nNon-example: $g(t) = e^{t^2}$ is of bounded variation on each $[0, T]$, but $\\int_0^T e^{-s t} \\, dg(t) = \\int_0^T 2 t e^{t^2 - s t} \\, dt$ diverges as $T \\to \\infty$ for every $s$, so no transform exists at any point.\nIn probability one often uses the Lebesgue–Stieltjes form $\\int_{[0, \\infty)} e^{-s t} \\, dG(t)$ for a distribution function $G$, which counts a possible jump at $t = 0$.\nWith $G(t) = 0$ for $t \u003c 0$ it equals $\\mathbb{E}[e^{-s X}]$ for $X \\ge 0$ with distribution function $G$.\nIf $g(0) = 0$ and $\\abs{g(t)} \\le M e^{a t}$ for all $t \\ge 0$, integration by parts gives $\\mathcal{L}^*\\{g\\}(s) = s \\, \\mathcal{L}\\{g\\}(s)$ for $\\operatorname{Re} s \u003e a$.\n",
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