{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]], and let $[0, \\infty]$ be the non-negative part of the [[extended real number line|0GSZ7B0CGR]].\nA *measure* on $(X, \\Sigma)$ is a function $\\mu \\colon \\Sigma \\to [0, \\infty]$ such that:\n\n1. $\\mu(\\varnothing) = 0$.\n2. $\\mu$ is [[countably additive|D7KB9HFCGZ]]: for every sequence $A_1, A_2, \\ldots$ of pairwise disjoint sets in $\\Sigma$,\n$$\\mu\\left(\\bigcup_{n=1}^{\\infty} A_n\\right) = \\sum_{n=1}^{\\infty} \\mu(A_n).$$\n\n## Remarks\n\nThe [[series|0NM626FH6Q]] in clause 2 has non-negative terms, so it has a value in $[0, \\infty]$.\nFor $x \\in X$, the [[Dirac measure|0ACRGYNHSR]] $\\delta_x$, with $\\delta_x(A) = 1$ if $x \\in A$ and $\\delta_x(A) = 0$ otherwise, is a measure.\nFor clause 2, the point $x$ lies in at most one of the disjoint sets $A_n$, so both sides are $1$ or both are $0$.\nOn the power set of $\\N$, the function with $\\nu(\\varnothing) = 0$ and $\\nu(A) = 1$ for $A \\ne \\varnothing$ is not a measure.\nClause 2 fails: $\\nu(\\set{1} \\cup \\set{2}) = 1$, while $\\nu(\\set{1}) + \\nu(\\set{2}) = 2$.\nA measure is monotone, $\\mu(A) \\le \\mu(B)$ when $A \\subseteq B$, and countably subadditive, $\\mu(\\bigcup_n A_n) \\le \\sum_n \\mu(A_n)$.\nA measure with $\\mu(X) = 1$ is a probability measure.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "PR11BZQQPG",
    "kind": "definition",
    "msc": [
      "28A12",
      "28A10"
    ],
    "name": "Measure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Measure_(mathematics)"
      }
    ],
    "tag": "7V9TSDJFZR",
    "uses": [
      "N4KDT83V3S",
      "0GSZ7B0CGR",
      "D7KB9HFCGZ",
      "0NM626FH6Q",
      "0ACRGYNHSR"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/7V9TSDJFZR/",
  "links": [
    {
      "field": "defines",
      "target": "PR11BZQQPG"
    },
    {
      "field": "uses",
      "target": "N4KDT83V3S"
    },
    {
      "field": "uses",
      "target": "0GSZ7B0CGR"
    },
    {
      "field": "uses",
      "target": "D7KB9HFCGZ"
    },
    {
      "field": "uses",
      "target": "0NM626FH6Q"
    },
    {
      "field": "uses",
      "target": "0ACRGYNHSR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7V9TSDJFZR.md",
  "msc": [
    "28A12",
    "28A10"
  ],
  "name": "Measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7V9/7V9TSDJFZR.md",
  "status": null,
  "tag": "7V9TSDJFZR",
  "uri": "m2://7V9TSDJFZR",
  "url": "https://math2.tamnd.com/wiki/measure-definition/",
  "writing": {
    "tag": "7V9TSDJFZR",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}