{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \\in \\R$.\nAssume that $f$ is [[measurable|YYNCJSR0PJ]] with respect to [[Lebesgue measure|2Z2QVWZ262]].\nThen there is a constant $c \\in \\R$ such that $f(x) = cx$ for all $x \\in \\R$, and in fact $c = f(1)$.\n\n## Remarks\n\nThe proof reduces the claim to boundedness on an interval.\nBy Lusin's theorem, $f$ is bounded on some compact set $K$ of positive measure.\nBy the Steinhaus theorem, the difference set $K - K$ contains an open interval around $0$.\nEach point $x = a - b$ of that interval with $a, b \\in K$ has $f(x) = f(a) - f(b)$, so $f$ is bounded on this interval.\nThen [[Darboux's theorem on additive functions|2F30ZHN1RT]] gives $f(x) = cx$.\nMeasurability cannot be dropped: [[a Hamel basis gives additive functions that are not linear|FFTNS66XG6]], and such functions are not Lebesgue measurable.\nThis is one of the regularity conditions that settle [[Cauchy's functional equation|ZBMEVWN100]].\n\n## Explanations\n\n### L1 human\n\nAn additive function turns sums into sums.\nThe only additive functions one can draw are the lines $f(x) = cx$ through the origin.\nThis theorem says that already a very weak regularity, measurability, forces an additive function to be such a line.\nSo the strange additive functions can never be measured or integrated.\n\n### L2 human\n\nMeasurability is weaker than continuity, monotonicity or local boundedness, so this result contains the other classical criteria.\nIn practice, any additive function that comes from an explicit formula, a limit of continuous functions or a probability model is measurable, so it is linear.\nThe same argument works for additive maps between $\\R^n$ and $\\R^m$.\nMore generally, a Haar measurable homomorphism between second countable locally compact groups is continuous.\nThe key tools, Lusin's theorem and the Steinhaus theorem, are standard in measure theory.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "KF719SZD5T"
    ],
    "kind": "statement",
    "msc": [
      "39B22",
      "28A20"
    ],
    "name": "Measurable solutions of Cauchy's equation",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation"
      }
    ],
    "tag": "TWANGP59RB",
    "uses": [
      "KF719SZD5T",
      "YYNCJSR0PJ",
      "2Z2QVWZ262"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/TWANGP59RB/",
  "links": [
    {
      "field": "about",
      "target": "KF719SZD5T"
    },
    {
      "field": "uses",
      "target": "KF719SZD5T"
    },
    {
      "field": "uses",
      "target": "YYNCJSR0PJ"
    },
    {
      "field": "uses",
      "target": "2Z2QVWZ262"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TWANGP59RB.md",
  "msc": [
    "39B22",
    "28A20"
  ],
  "name": "Measurable solutions of Cauchy's equation",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TWA/TWANGP59RB.md",
  "status": null,
  "tag": "TWANGP59RB",
  "uri": "m2://TWANGP59RB",
  "url": "https://tamnd.github.io/math2/wiki/measurable-solutions-of-cauchys-equation/",
  "writing": {
    "tag": "TWANGP59RB",
    "kind": "statement",
    "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
      }
    ]
  }
}