{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nLet $f \\colon X \\to \\C$ or $f \\colon X \\to [-\\infty, \\infty]$ be a [[measurable function|YYNCJSR0PJ]].\nThe *distribution function* of $f$ is the function $d_f \\colon [0, \\infty) \\to [0, \\infty]$ given by\n$$d_f(t) = \\mu(\\set{x \\in X : \\abs{f(x)} \u003e t}).$$\n\n## Remarks\n\nExample: let $X = [0, 1]$ with Lebesgue measure and $f(x) = x^2$.\nFor $0 \\le t \u003c 1$ the set $\\set{x : x^2 \u003e t}$ is $(\\sqrt{t}, 1]$, so $d_f(t) = 1 - \\sqrt{t}$.\nFor $t \\ge 1$ the set is empty, so $d_f(t) = 0$.\n\nNon-example: the cumulative distribution function $F(t) = \\mu(\\set{f \\le t})$ of probability theory is a different object.\nIt measures where $f$ is small, uses $f$ and not $\\abs{f}$, and is increasing rather than decreasing.\n\nThe function $d_f$ is non-increasing and right-continuous.\nTwo functions with the same distribution function are called *equimeasurable*, and they have the same $L^p$ norms.\nSome authors write $\\lambda_f$ or $\\mu_f$ for $d_f$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "HQVXKEMN5N",
    "kind": "definition",
    "msc": [
      "28A20",
      "46E30"
    ],
    "name": "Distribution function of a measurable function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Layer_cake_representation"
      }
    ],
    "tag": "NB0HE0TE1R",
    "uses": [
      "P15VQS5DJ9",
      "YYNCJSR0PJ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/NB0HE0TE1R/",
  "links": [
    {
      "field": "defines",
      "target": "HQVXKEMN5N"
    },
    {
      "field": "uses",
      "target": "P15VQS5DJ9"
    },
    {
      "field": "uses",
      "target": "YYNCJSR0PJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NB0HE0TE1R.md",
  "msc": [
    "28A20",
    "46E30"
  ],
  "name": "Distribution function of a measurable function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NB0/NB0HE0TE1R.md",
  "status": null,
  "tag": "NB0HE0TE1R",
  "uri": "m2://NB0HE0TE1R",
  "url": "https://math2.tamnd.com/wiki/distribution-function-of-a-measurable-function-definition/",
  "writing": {
    "tag": "NB0HE0TE1R",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}