{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $g \\colon \\R \\to \\R$ be a [[non-decreasing|0WFF6AZPBP]], right-continuous function.\nLet $\\mu_g$ be its [[Lebesgue–Stieltjes measure|QV75XJBFCS]], the Borel measure with $\\mu_g((a, b]) = g(b) - g(a)$ for all $a \u003c b$.\nLet $f \\colon \\R \\to \\R$ be a Borel measurable function that is [[integrable|X7GTF3D98F]] with respect to $\\mu_g$.\nThe *Lebesgue–Stieltjes integral* of $f$ with respect to $g$ is the [[Lebesgue integral|DWJF7ABPR1]]\n$$\\int_\\R f \\, dg = \\int_\\R f \\, d\\mu_g.$$\n\n## Remarks\n\nExample: let $g = \\mathbf{1}_{[0, \\infty)}$, the Heaviside step function with $g(0) = 1$, which is non-decreasing and right-continuous.\nThen $\\mu_g$ is the Dirac measure at $0$, so $\\int f \\, dg = f(0)$ for every Borel function $f$.\nFor $g(x) = x$ the measure $\\mu_g$ is Lebesgue measure, and the integral is the usual Lebesgue integral.\n\nNon-example: $g(x) = -x$ is not non-decreasing, so the definition does not apply directly.\nThe formula $\\mu_g((a, b]) = g(b) - g(a)$ would give negative mass.\n\nFor $g$ of bounded variation and right-continuous, one writes $g = g_1 - g_2$ with $g_1, g_2$ non-decreasing and right-continuous, and sets $\\int f \\, dg = \\int f \\, dg_1 - \\int f \\, dg_2$.\nOn $[a, b]$ one writes $\\int_{(a, b]} f \\, dg$, and the endpoints matter when $g$ jumps there.\nIf $f$ is continuous on $[a, b]$, the integral over $(a, b]$ equals the Riemann–Stieltjes integral $\\int_a^b f \\, dg$.\nIf $g$ is absolutely continuous on bounded intervals, then $\\int f \\, dg = \\int f(x) g'(x) \\, dx$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "FK3QBZAZH7",
    "kind": "definition",
    "msc": [
      "26A42",
      "28A25"
    ],
    "name": "Lebesgue–Stieltjes integration",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration"
      }
    ],
    "tag": "CC8KNGMZYS",
    "uses": [
      "0WFF6AZPBP",
      "QV75XJBFCS",
      "X7GTF3D98F",
      "DWJF7ABPR1"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CC8KNGMZYS/",
  "links": [
    {
      "field": "defines",
      "target": "FK3QBZAZH7"
    },
    {
      "field": "uses",
      "target": "0WFF6AZPBP"
    },
    {
      "field": "uses",
      "target": "QV75XJBFCS"
    },
    {
      "field": "uses",
      "target": "X7GTF3D98F"
    },
    {
      "field": "uses",
      "target": "DWJF7ABPR1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CC8KNGMZYS.md",
  "msc": [
    "26A42",
    "28A25"
  ],
  "name": "Lebesgue–Stieltjes integration",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CC8/CC8KNGMZYS.md",
  "status": null,
  "tag": "CC8KNGMZYS",
  "uri": "m2://CC8KNGMZYS",
  "url": "https://math2.tamnd.com/wiki/lebesgue-stieltjes-integration-definition/",
  "writing": {
    "tag": "CC8KNGMZYS",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}