{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5DHPVQ9B4X"
    },
    {
      "field": "uses",
      "target": "ATG4DQZZT9"
    },
    {
      "field": "uses",
      "target": "B6RYM7QTYZ"
    },
    {
      "field": "uses",
      "target": "GEA8H3228S"
    },
    {
      "field": "about",
      "target": "B6RYM7QTYZ"
    },
    {
      "field": "defines",
      "target": "QB6EZ3HP73"
    }
  ],
  "body": "\n## Summary\n\nA simple function is a [[measurable function|YYNCJSR0PJ]] that takes only finitely many values.\nIt is a finite combination of [[indicator functions|0RCQZ36MME]] of measurable sets, so its integral is a finite sum.\nThe Lebesgue integral of a nonnegative function is defined by approximating it from below by simple functions.\n\n## Explanations\n\n### L1 human\n\nA simple function is like a staircase, except that each step may sit on a complicated set instead of an interval.\nOn each set it has a constant value, and there are only finitely many sets.\nFor example, the function on $[0, 1]$ that is $2$ on the rational numbers and $5$ on the irrational numbers is simple.\nIts integral is $2$ times the measure of the rationals plus $5$ times the measure of the irrationals, which is $0 + 5 = 5$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28A20"
    ],
    "name": "Simple function",
    "same_as": {
      "wikidata": "Q913022",
      "wikipedia": "Simple function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Simple_function"
      }
    ],
    "specializes": [
      "YYNCJSR0PJ"
    ],
    "tag": "7WMBX7M4P6"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/7WMBX7M4P6/",
  "links": [
    {
      "field": "specializes",
      "target": "YYNCJSR0PJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7WMBX7M4P6.md",
  "msc": [
    "28A20"
  ],
  "name": "Simple function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7WM/7WMBX7M4P6.md",
  "status": null,
  "tag": "7WMBX7M4P6",
  "uri": "m2://7WMBX7M4P6",
  "url": "https://math2.tamnd.com/wiki/simple-function/",
  "writing": {
    "tag": "7WMBX7M4P6",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}