{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "17CW7Z2YDT"
    },
    {
      "field": "uses",
      "target": "64F9VMG27D"
    },
    {
      "field": "uses",
      "target": "CM6XF84MNY"
    },
    {
      "field": "uses",
      "target": "JXK6NRDH4V"
    },
    {
      "field": "uses",
      "target": "R80G20B2ZW"
    }
  ],
  "body": "\n## Summary\n\nA finitely additive measure is a function on an algebra of sets with values in $[0, \\infty]$.\nIt is $0$ on the empty set and adds over finite disjoint unions.\nIt is weaker than a measure, which must add over countable disjoint unions.\nFinitely additive measures appear in the construction of measures, in densities on the integers and in the duals of spaces of bounded functions.\n\n## Explanations\n\n### L1 human\n\nA finitely additive measure gives sizes to sets so that the size of two separate pieces together is the sum of their sizes.\nNothing is required for infinitely many pieces.\nFor example, on the natural numbers, give size $0$ to finite sets and size $1$ to sets with finite complement.\nTwo disjoint pieces cannot both have finite complement, so sizes add.\nBut $\\N$ is the union of the singletons $\\set{n}$, each of size $0$, so the rule fails for infinitely many pieces.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28A12",
      "28A10"
    ],
    "name": "Finitely additive measure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Content_(measure_theory)"
      }
    ],
    "tag": "KFWQ24D9M6"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/KFWQ24D9M6/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/KFWQ24D9M6.md",
  "msc": [
    "28A12",
    "28A10"
  ],
  "name": "Finitely additive measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KFW/KFWQ24D9M6.md",
  "status": null,
  "tag": "KFWQ24D9M6",
  "uri": "m2://KFWQ24D9M6",
  "url": "https://math2.tamnd.com/wiki/finitely-additive-measure/",
  "writing": {
    "tag": "KFWQ24D9M6",
    "kind": "concept",
    "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
      }
    ]
  }
}