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  "body": "\n## Summary\n\nThe Laplace–Stieltjes transform of a function $g$ on $[0, \\infty)$ is $\\int_0^\\infty e^{-s t} \\, dg(t)$, a [[Riemann–Stieltjes integral|JT9AFHG0NA]] against $g$ instead of an ordinary integral of $g$.\nIt extends the [[Laplace transform|0V49JEZW06]]: when $g$ has a density $f$, it is the Laplace transform of $f$.\nIt also handles jumps of $g$, that is, point masses.\nIn probability, applied to the distribution function of a nonnegative random variable $X$, it gives $\\mathbb{E}[e^{-s X}]$.\n\n## Explanations\n\n### L1 human\n\nThink of $g$ as describing how mass is laid out along the half-line: $g(t)$ is the total mass in $[0, t]$.\nThe transform weights each bit of mass at position $t$ by $e^{-s t}$ and adds up.\nA single point mass at $a$ gives just $e^{-a s}$.\nA uniform spread of mass with density $1$ gives $1/s$, the ordinary Laplace transform of the constant $1$.\n",
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