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  "body": "\n## Definition\n\nLet $0 \\le m \\le n$, and let $T$ be an $m$-dimensional [[current|360H7MD9RD]] on $\\R^n$, with mass $\\mathbf{M}(T)$ and boundary $\\partial T$.\nThe current $T$ is *integer multiplicity rectifiable* if there are:\n\n1. an $\\mathcal{H}^m$-measurable countably $m$-[[rectifiable set|7N0GFMW05R]] $S \\subseteq \\R^n$, where $\\mathcal{H}^m$ is the [[Hausdorff measure|Q1TGZK8RN6]];\n2. a locally $\\mathcal{H}^m$-integrable function $\\theta \\colon S \\to \\set{1, 2, 3, \\dots}$;\n3. an $\\mathcal{H}^m$-measurable map $\\xi$ that assigns to $\\mathcal{H}^m$-almost every $x \\in S$ a unit simple $m$-vector $\\xi(x)$ orienting the approximate tangent space of $S$ at $x$;\n\nsuch that $T(\\omega) = \\int_S \\ip{\\omega(x)}{\\xi(x)} \\, \\theta(x) \\, d\\mathcal{H}^m(x)$ for every smooth compactly supported $m$-form $\\omega$.\nThe current $T$ is an *integral current* if it is integer multiplicity rectifiable, has compact support, satisfies $\\mathbf{M}(T) \u003c \\infty$, and, when $m \\ge 1$, satisfies $\\mathbf{M}(\\partial T) \u003c \\infty$.\n\n## Remarks\n\nBy the boundary rectifiability theorem of Federer and Fleming, the boundary of an integral current is again an integral current.\nFederer's original definition asks instead that $T$ and $\\partial T$ be rectifiable currents, and the two definitions agree by that theorem.\nThe integral $0$-currents are the finite sums $\\sum_i a_i \\delta_{p_i}$ with $a_i \\in \\Z$.\n\nExample: let $D$ be the closed unit disc in $\\R^2 \\subseteq \\R^3$, oriented by $e_1 \\wedge e_2$.\nThen $[\\![D]\\!](\\omega) = \\int_D \\omega$ is an integral $2$-current: take $S = D$, $\\theta = 1$, $\\xi = e_1 \\wedge e_2$.\nIts mass is $\\pi$, and its boundary is the circle, of mass $2\\pi$.\n\nNon-example: in $\\R$, let $T = \\sum_{k \\ge 1} [\\![I_k]\\!]$, where $I_k = [1/(2k + 1), 1/(2k)]$ is oriented by $e_1$.\nIt is integer multiplicity rectifiable with compact support in $[0, 1]$ and $\\mathbf{M}(T) = \\sum_k \\frac{1}{2k(2k + 1)} \u003c \\infty$.\nBut $\\partial T$ is a sum of infinitely many unit point masses, so $\\mathbf{M}(\\partial T) = \\infty$ and $T$ is not an integral current.\n\n## Explanations\n\n### L1 human\n\nAn integral current is integration over an oriented rectifiable set with positive integer multiplicities, with finite area and finite boundary length.\n",
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