{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a function.\nA *gauge* on $[a, b]$ is a function $\\delta \\colon [a, b] \\to (0, \\infty)$.\nA tagged [[partition|069P20KYQT]] $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_n = b$ with tags $t_i \\in [x_{i-1}, x_i]$ is *$\\delta$-fine* if $[x_{i-1}, x_i] \\subseteq (t_i - \\delta(t_i), t_i + \\delta(t_i))$ for every $i$.\nThe function $f$ is *Henstock–Kurzweil integrable* on $[a, b]$, with integral $A \\in \\R$, if the following holds.\nFor every $\\varepsilon \u003e 0$ there is a gauge $\\delta$ such that\n$$\\Bigl\\lvert \\sum_{i=1}^{n} f(t_i)(x_i - x_{i-1}) - A \\Bigr\\rvert \u003c \\varepsilon$$\nfor every $\\delta$-fine tagged partition of $[a, b]$.\n\n## Remarks\n\nCousin's lemma says that for every gauge a $\\delta$-fine tagged partition exists, so the integral, when it exists, is unique.\nIf the gauges can always be taken constant, the definition is that of the [[Riemann integral|0QM4TXT3DR]].\n\nExample: let $F(x) = x^2 \\sin(1/x^2)$ for $x \\ne 0$ and $F(0) = 0$, and $f = F'$ on $[0, 1]$.\nThe function $F$ is differentiable everywhere, so $f$ is Henstock–Kurzweil integrable with integral $F(1) - F(0) = \\sin 1$.\nHere $f(x) = 2x \\sin(1/x^2) - (2/x)\\cos(1/x^2)$ for $x \\ne 0$, and $\\abs{f}$ is not Lebesgue integrable.\n\nNon-example: $f(x) = 1/x$ for $0 \u003c x \\le 1$ and $f(0) = 0$.\nA nonnegative Henstock–Kurzweil integrable function is Lebesgue integrable, and $\\int_0^1 dx/x = \\infty$, so $f$ is not integrable.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "FCGHXDGVV9",
    "kind": "definition",
    "msc": [
      "26A39"
    ],
    "name": "Henstock–Kurzweil integral",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral"
      }
    ],
    "tag": "Y3KYKTAMCT",
    "uses": [
      "069P20KYQT",
      "0HBY48G8G1"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Y3KYKTAMCT/",
  "links": [
    {
      "field": "defines",
      "target": "FCGHXDGVV9"
    },
    {
      "field": "uses",
      "target": "069P20KYQT"
    },
    {
      "field": "uses",
      "target": "0HBY48G8G1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y3KYKTAMCT.md",
  "msc": [
    "26A39"
  ],
  "name": "Henstock–Kurzweil integral",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y3K/Y3KYKTAMCT.md",
  "status": null,
  "tag": "Y3KYKTAMCT",
  "uri": "m2://Y3KYKTAMCT",
  "url": "https://math2.tamnd.com/wiki/henstock-kurzweil-integral-definition/",
  "writing": {
    "tag": "Y3KYKTAMCT",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}