{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $0 \\le m \\le n$, let $K \\subseteq \\R^n$ be compact, and let $C \u003e 0$.\nLet $(T_j)_{j \\ge 1}$ be a sequence of [[integral currents|EG4AJVEFT2]] of dimension $m$ on $\\R^n$ with support in $K$.\nAssume $\\mathbf{M}(T_j) + \\mathbf{M}(\\partial T_j) \\le C$ for all $j$, where $\\mathbf{M}$ is the mass and $\\partial T_j = 0$ when $m = 0$.\nThen there are a subsequence $(T_{j_i})$ and an integral $m$-current $T$ with support in $K$ such that $\\mathbf{F}(T_{j_i} - T) \\to 0$, where $\\mathbf{F}$ is the [[flat norm|Y7P66D41Z8]].\nMoreover $\\mathbf{M}(T) \\le \\liminf_i \\mathbf{M}(T_{j_i})$ and $\\mathbf{M}(\\partial T) \\le \\liminf_i \\mathbf{M}(\\partial T_{j_i})$.\n\n## Remarks\n\nWeak convergence of a subsequence follows from the mass bound by a diagonal argument on a countable dense set of forms.\nThe mass is lower semicontinuous under weak convergence, because it is a supremum of the continuous functionals $T \\mapsto T(\\omega)$.\nThe hard part is the closure statement: the weak limit is again integral, and the convergence holds in the flat norm.\nFederer and Fleming proved it with the deformation theorem, which pushes currents onto a cubical grid; later proofs use slicing.\n\nExample at work: let $B$ be an integral $(m-1)$-current with compact support that is the boundary of some integral $m$-current.\nA minimising sequence for the mass among integral currents with boundary $B$ can be projected into a fixed ball without increasing mass, and its boundary masses are all $\\mathbf{M}(B)$.\nThe theorem gives a limit with boundary $B$ and least mass, an [[area-minimizing current|4NKP73KVVW]].\n\nThe bound on $\\mathbf{M}(\\partial T_j)$ is needed.\nIn $\\R$, let $T_j = \\sum_{i=0}^{j-1} [\\![i/j, \\, i/j + 1/(2j)]\\!]$, integral $1$-currents in $[0, 1]$ with mass $\\frac{1}{2}$ and $\\mathbf{M}(\\partial T_j) = 2j$.\nThey converge weakly to $\\frac{1}{2} [\\![0, 1]\\!]$, which has multiplicity $\\frac{1}{2}$ and is not an integral current.\n\n## Explanations\n\n### L1 human\n\nGeneralised surfaces with integer multiplicities, with bounded area and bounded boundary length, inside a fixed bounded region, always have a convergent subsequence.\nThe limit is again such a surface, and area can only drop in the limit.\nThis is what makes it possible to find surfaces of least area by taking limits of better and better ones.\n\n### L2 human\n\nThe theorem is the compactness step of the direct method of the calculus of variations for currents.\nCombined with lower semicontinuity of mass, it solves the Plateau problem for integral currents in every dimension and codimension.\nRegularity theory, such as the results of De Giorgi, Almgren and Allard, then shows that the minimisers are smooth away from a small singular set.\nThe same compactness, with the flat norm, underlies the theory of flat chains and of minimal surfaces obtained by min-max methods.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "EG4AJVEFT2"
    ],
    "kind": "statement",
    "msc": [
      "49Q15",
      "28A75",
      "49Q20"
    ],
    "name": "Federer–Fleming compactness theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Current_(mathematics)"
      }
    ],
    "tag": "TPXYARH912",
    "uses": [
      "EG4AJVEFT2",
      "Y7P66D41Z8",
      "360H7MD9RD",
      "4NKP73KVVW"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/TPXYARH912/",
  "links": [
    {
      "field": "about",
      "target": "EG4AJVEFT2"
    },
    {
      "field": "uses",
      "target": "EG4AJVEFT2"
    },
    {
      "field": "uses",
      "target": "Y7P66D41Z8"
    },
    {
      "field": "uses",
      "target": "360H7MD9RD"
    },
    {
      "field": "uses",
      "target": "4NKP73KVVW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TPXYARH912.md",
  "msc": [
    "49Q15",
    "28A75",
    "49Q20"
  ],
  "name": "Federer–Fleming compactness theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TPX/TPXYARH912.md",
  "status": null,
  "tag": "TPXYARH912",
  "uri": "m2://TPXYARH912",
  "url": "https://tamnd.github.io/math2/wiki/federer-fleming-compactness-theorem/",
  "writing": {
    "tag": "TPXYARH912",
    "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
      }
    ]
  }
}