{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a topological space and let $\\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.\nA Borel set $A \\subseteq X$ is *outer regular* for $\\mu$ if\n$$\\mu(A) = \\inf \\set{\\mu(U) : U \\supseteq A, \\ U \\text{ open}}.$$\nThe measure $\\mu$ is *outer regular* if every Borel set is outer regular for $\\mu$.\n\n## Remarks\n\nEvery [[open set|02BH3MCBQB]] is outer regular, by taking $U = A$.\n\nExample: Lebesgue measure $\\lambda$ on the Borel sets of $\\R$ is outer regular.\nFor a Borel set $A$ and $\\varepsilon \u003e 0$, the outer measure construction gives intervals $I_k$ covering $A$ with $\\sum_k \\lambda(I_k) \u003c \\lambda(A) + \\varepsilon$.\nEnlarging each $I_k$ slightly to an open interval gives an open $U \\supseteq A$ with $\\lambda(U) \u003c \\lambda(A) + 2\\varepsilon$.\n\nNon-example: counting measure $\\mu$ on $\\R$ is not outer regular.\nThe set $\\{0\\}$ has $\\mu(\\{0\\}) = 1$, but every open set containing $0$ is infinite and has measure $\\infty$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "33C1KEK3SC",
    "kind": "definition",
    "msc": [
      "28C15"
    ],
    "name": "Outer regular measure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Radon_measure"
      }
    ],
    "tag": "J6MTQSZVEM",
    "uses": [
      "DYR1QFNGNJ",
      "02BH3MCBQB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/J6MTQSZVEM/",
  "links": [
    {
      "field": "defines",
      "target": "33C1KEK3SC"
    },
    {
      "field": "uses",
      "target": "DYR1QFNGNJ"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/J6MTQSZVEM.md",
  "msc": [
    "28C15"
  ],
  "name": "Outer regular measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J6M/J6MTQSZVEM.md",
  "status": null,
  "tag": "J6MTQSZVEM",
  "uri": "m2://J6MTQSZVEM",
  "url": "https://math2.tamnd.com/wiki/outer-regular-measure-definition/",
  "writing": {
    "tag": "J6MTQSZVEM",
    "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
      }
    ]
  }
}