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  "body": "\n## Summary\n\nThe Riemann–Stieltjes integral $\\int_a^b f \\, dg$ integrates a function $f$ against another function $g$, the integrator.\nIts sums weight each value of $f$ by the change of $g$ over a subinterval instead of by the length of the subinterval.\nWith $g(x) = x$ it is the [[Riemann integral|0QM4TXT3DR]].\nIt handles sums and integrals in one formula.\nIt is the setting of the Riesz representation of continuous linear functionals on $C[a, b]$, and of expectations written with a [[cumulative distribution function|0MXMS3AE13]].\n\n## Explanations\n\n### L1 human\n\nIn an ordinary integral every piece of the $x$-axis counts by its length.\nIn a Stieltjes integral a piece counts by how much $g$ grows on it, so $g$ decides where the mass sits.\nIf $g$ jumps by $1$ at the point $c$ and is constant elsewhere, then $\\int_a^b f \\, dg = f(c)$ for continuous $f$: the integral just reads off a value.\nIf $g$ is a step function with several jumps, the integral becomes a finite sum.\n",
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