{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "C3CKZAZ57Z"
    },
    {
      "field": "about",
      "target": "Q3Y1HBD8NN"
    },
    {
      "field": "uses",
      "target": "C3CKZAZ57Z"
    },
    {
      "field": "uses",
      "target": "Q3Y1HBD8NN"
    },
    {
      "field": "defines",
      "target": "FR210Q35CB"
    }
  ],
  "body": "\n## Summary\n\nA family of integrable functions is uniformly integrable when the part of each function above a large level carries uniformly little integral.\nIt prevents mass from escaping into tall spikes, in the same way for every member of the family.\nOn a finite measure space it is exactly the condition that upgrades convergence almost everywhere or in measure to convergence in $L^1$.\nIt plays a central role in probability, for instance in martingale convergence.\n\n## Explanations\n\n### L1 human\n\nA single integrable function puts only a small part of its integral on the set where it is very large.\nUniform integrability asks this for a whole family at once, with the same threshold for all members.\nFor example, all functions bounded by the constant $1$ on $[0, 1]$ form a uniformly integrable family.\nThe spikes $n \\mathbf{1}_{(0, 1/n)}$ do not, since each one puts all of its integral $1$ above the level $n - 1$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28A25",
      "60A10"
    ],
    "name": "Uniform integrability",
    "same_as": {
      "wikidata": "Q1530275",
      "wikipedia": "Uniform integrability"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Uniform_integrability"
      }
    ],
    "tag": "VRVVKMXDGV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/VRVVKMXDGV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/VRVVKMXDGV.md",
  "msc": [
    "28A25",
    "60A10"
  ],
  "name": "Uniform integrability",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VRV/VRVVKMXDGV.md",
  "status": null,
  "tag": "VRVVKMXDGV",
  "uri": "m2://VRVVKMXDGV",
  "url": "https://math2.tamnd.com/wiki/uniform-integrability/",
  "writing": {
    "tag": "VRVVKMXDGV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}