{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set and let $H$ be a set of functions $X \\to \\R$ with the following property.\n$H$ is a [[vector space|05EGHEQZAW]] under pointwise operations, and $\\max(h, 0) \\in H$ for every $h \\in H$.\nA *Daniell integral*, or elementary integral, on $H$ is a map $I \\colon H \\to \\R$ such that:\n\n1. $I$ is linear.\n2. $I$ is positive: $I(h) \\ge 0$ for every $h \\in H$ with $h \\ge 0$.\n3. If $h_1 \\ge h_2 \\ge \\cdots$ in $H$ and $h_n \\to 0$ [[pointwise|056FSKG32J]] on $X$, then $I(h_n) \\to 0$.\n\n## Remarks\n\nExample: let $H$ be the continuous functions $\\R \\to \\R$ with compact support, and $I(h) = \\int h(x) \\, dx$, the Riemann integral.\nLinearity and positivity are clear.\nFor clause 3, all $h_n$ vanish outside a compact interval $K$ that contains the support of $h_1$.\nBy Dini's theorem $h_n \\to 0$ uniformly on $K$, so $0 \\le I(h_n) \\le \\max h_n \\cdot \\operatorname{length}(K) \\to 0$.\n\nNon-example: let $H$ be the convergent real sequences on $X = \\N$, and $I(h) = \\lim_{n \\to \\infty} h(n)$.\nClauses 1 and 2 hold.\nClause 3 fails: $h_n = \\mathbf{1}_{\\set{n, n+1, \\dots}}$ decreases to $0$ pointwise, but $I(h_n) = 1$ for all $n$.\n\nDaniell's construction extends $I$ to a class of integrable functions on which the monotone and dominated convergence theorems hold.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "4KCBFQS9WR",
    "kind": "definition",
    "msc": [
      "28C05",
      "28A25"
    ],
    "name": "Daniell integral",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Daniell_integral"
      }
    ],
    "tag": "3RQ2PT36KQ",
    "uses": [
      "05EGHEQZAW",
      "056FSKG32J"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/3RQ2PT36KQ/",
  "links": [
    {
      "field": "defines",
      "target": "4KCBFQS9WR"
    },
    {
      "field": "uses",
      "target": "05EGHEQZAW"
    },
    {
      "field": "uses",
      "target": "056FSKG32J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3RQ2PT36KQ.md",
  "msc": [
    "28C05",
    "28A25"
  ],
  "name": "Daniell integral",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3RQ/3RQ2PT36KQ.md",
  "status": null,
  "tag": "3RQ2PT36KQ",
  "uri": "m2://3RQ2PT36KQ",
  "url": "https://math2.tamnd.com/wiki/daniell-integral-definition/",
  "writing": {
    "tag": "3RQ2PT36KQ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}