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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f, g \\colon [a, b] \\to \\R$ be functions.\nA tagged [[partition|069P20KYQT]] $(P, c)$ of $[a, b]$ consists of points $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_n = b$ and tags $c_i \\in [x_{i-1}, x_i]$.\nIts mesh is $\\max_i (x_i - x_{i-1})$, and its Riemann–Stieltjes sum is\n$$S(P, c, f, g) = \\sum_{i=1}^{n} f(c_i) \\bigl(g(x_i) - g(x_{i-1})\\bigr).$$\nThe function $f$ is *Riemann–Stieltjes integrable* with respect to $g$, with integral $A \\in \\R$, if the following holds.\nFor every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $\\abs{S(P, c, f, g) - A} \u003c \\varepsilon$ for every tagged partition $(P, c)$ of mesh less than $\\delta$.\nOne writes $A = \\int_a^b f(x)\\, dg(x)$.\n\n## Remarks\n\nExample: if $f$ is continuous and $g$ is continuously differentiable on $[a, b]$, then $\\int_a^b f \\, dg = \\int_a^b f(x) g'(x)\\, dx$.\nBy the mean value theorem, $g(x_i) - g(x_{i-1}) = g'(d_i)(x_i - x_{i-1})$ for some $d_i \\in (x_{i-1}, x_i)$.\nUniform continuity of $g'$ lets one replace $d_i$ by $c_i$ with a small error, which leaves a [[Riemann sum|0HBY48G8G1]] of $f g'$.\nWith $g(x) = x$ the definition is that of the [[Riemann integral|0QM4TXT3DR]].\n\nNon-example: on $[-1, 1]$ let $f = g$ be $0$ on $[-1, 0]$ and $1$ on $(0, 1]$.\nIf $0 = x_k$ is a partition point, the only nonzero term is $f(c_{k+1})$.\nIt is $0$ for the tag $c_{k+1} = 0$ and $1$ for any other tag.\nSo sums of every small mesh take both values $0$ and $1$, and $f$ is not integrable with respect to $g$.\n\nIf $f$ is continuous and $g$ has [[bounded variation|A8251BHCMP]], then $\\int_a^b f \\, dg$ exists.\nSome authors use the limit over refinements of partitions instead of the mesh, which gives a slightly larger class of integrable pairs.\n",
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