{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *measure algebra* is a pair $(\\mathfrak{A}, \\mu)$ such that:\n\n1. $\\mathfrak{A}$ is a [[Boolean algebra|0XADJKM699]] in which every countable subset has a supremum.\n2. $\\mu \\colon \\mathfrak{A} \\to [0, \\infty]$ satisfies $\\mu(0) = 0$.\n3. $\\mu\\left(\\bigvee_{k} a_k\\right) = \\sum_k \\mu(a_k)$ for every sequence $(a_k)_{k \\in \\N}$ of pairwise disjoint elements, that is, $a_j \\wedge a_k = 0$ for $j \\ne k$.\n4. $\\mu(a) \u003e 0$ for every $a \\ne 0$.\n\n## Remarks\n\nExample: let $(X, \\Sigma, \\nu)$ be a [[measure space|P15VQS5DJ9]] and $\\mathcal{N} = \\set{E \\in \\Sigma : \\nu(E) = 0}$.\nCall $E, F \\in \\Sigma$ equivalent when $\\nu(E \\mathbin{\\triangle} F) = 0$.\nThe classes form a Boolean algebra $\\Sigma / \\mathcal{N}$ with countable suprema given by countable unions, and $\\mu([E]) = \\nu(E)$ is well defined.\nClauses 2 and 3 come from the measure $\\nu$, and clause 4 holds since $[E] \\ne 0$ means $\\nu(E) \u003e 0$.\n\nNon-example: the [[sigma-algebra|0SV8K0TDGY]] of Borel subsets of $\\R$ with Lebesgue measure satisfies clauses 1 to 3 but not 4.\nThe set $\\set{0}$ is a nonzero element of measure $0$.\n\nThe quotient in the example is called the measure algebra of $(X, \\Sigma, \\nu)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "YZA1V04R96",
    "kind": "definition",
    "msc": [
      "28A60",
      "06E10"
    ],
    "name": "Measure algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Measure_algebra"
      }
    ],
    "tag": "KST0YNAP55",
    "uses": [
      "0XADJKM699",
      "PR11BZQQPG"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/KST0YNAP55/",
  "links": [
    {
      "field": "defines",
      "target": "YZA1V04R96"
    },
    {
      "field": "uses",
      "target": "0XADJKM699"
    },
    {
      "field": "uses",
      "target": "PR11BZQQPG"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KST0YNAP55.md",
  "msc": [
    "28A60",
    "06E10"
  ],
  "name": "Measure algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KST/KST0YNAP55.md",
  "status": null,
  "tag": "KST0YNAP55",
  "uri": "m2://KST0YNAP55",
  "url": "https://tamnd.github.io/math2/wiki/measure-algebra-definition/",
  "writing": {
    "tag": "KST0YNAP55",
    "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
      }
    ]
  }
}