{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "E77T9G4DV7"
    }
  ],
  "body": "\n## Summary\n\nLittlewood's three principles are rules of thumb for working with [[Lebesgue measure|2Z2QVWZ262]] on the real line.\nEvery measurable set is nearly a finite union of intervals, and every [[measurable function|YYNCJSR0PJ]] is nearly continuous.\nEvery convergent sequence of measurable functions is nearly uniformly convergent.\nEach principle has a precise form, and the last two are [[Lusin's theorem|GZMAH9W2SK]] and [[Egorov's theorem|762Y2NYA7W]].\nThey let one transfer arguments for simple objects to general measurable ones.\n\n## Explanations\n\n### L1 human\n\nMeasurable sets and functions can look wild, but up to a small error they behave like familiar ones.\nThe word \"nearly\" always means: after throwing away a set of arbitrarily small measure.\nFor example, the set of rational numbers in $[0, 1]$ looks scattered everywhere, but its measure is $0$, so it is nearly the empty set.\nIts indicator function is discontinuous everywhere, yet it is nearly the zero function, which is continuous.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "28A20",
      "26A42"
    ],
    "name": "Littlewood's three principles",
    "same_as": {
      "wikidata": "Q6653163",
      "wikipedia": "Littlewood's three principles of real analysis"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Littlewood%27s_three_principles_of_real_analysis"
      }
    ],
    "tag": "JXFHBY77RN"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/JXFHBY77RN/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/JXFHBY77RN.md",
  "msc": [
    "28A20",
    "26A42"
  ],
  "name": "Littlewood's three principles",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JXF/JXFHBY77RN.md",
  "status": null,
  "tag": "JXFHBY77RN",
  "uri": "m2://JXFHBY77RN",
  "url": "https://math2.tamnd.com/wiki/littlewoods-three-principles/",
  "writing": {
    "tag": "JXFHBY77RN",
    "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
      }
    ]
  }
}